File "hl-mixed-differentiation.html"
Path: /StudyIB/mathsanalysis/page/2405/hl-mixed-differentiationhtml
File size: 206.28 KB
MIME-type: text/html
Charset: utf-8
<!DOCTYPE html><html lang="EN"><head> <script>(function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start': new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0], j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src= 'https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f); })(window,document,'script','dataLayer','GTM-PZBWZ8J');</script> <title>HL Mixed Differentiation</title><!-- Removed by WebCopy --><!--<base href="/">--><!-- Removed by WebCopy --><meta http-equiv="x-ua-compatible" content="IE=Edge"><meta charset="utf-8"><meta name="viewport" content="width=device-width, initial-scale=1"><link href="../../../css/bootstrap.flatly.min.css" rel="stylesheet" media="screen"><link rel="stylesheet" href="../../../fonts/awesome/css/font-awesome.min.css"><link href="../../../js/jquery-fancybox/jquery.fancybox.min.css" type="text/css" rel="stylesheet"><link rel="stylesheet" href="../../../css/style.min.css?v=202301301645"><meta name="robots" content="index, follow"><meta name="googlebot" content="noarchive"><meta prefix="og: http://ogp.me/ns#" property="og:title" name="og:title" content="StudyIB Maths: Analysis & Approaches: HL Mixed Differentiation"> <meta prefix="og: http://ogp.me/ns#" property="og:image" content="https://studyib.net/img/studyib-card-default.jpg"> <meta prefix="og: http://ogp.me/ns#" property="og:description" name="og:description" content="This page is ideal for practising all the skills of differentiation. You may wish to use this page in preparation for a test on this topic or for the final examinations. The quizzes on this page have been carefully created to take you through..."> <meta prefix="og: http://ogp.me/ns#" property="og:url" name="og:url" content="https://studyib.net/mathsanalysis/page/2405/hl-mixed-differentiation"> <meta prefix="og: http://ogp.me/ns#" property="og:site_name" name="og:site_name" content="StudyIB - Your IB learning companion"> <meta prefix="og: http://ogp.me/ns#" property="og:locale" name="og:locale" content="en"> <meta prefix="og: http://ogp.me/ns#" property="og:type" name="og:type" content="website"> <meta name="description" content="This page is ideal for practising all the skills of differentiation. You may wish to use this page in preparation for a test on this topic or for the final examinations. The quizzes on this page have been carefully created to take you through..."> <meta name="image" content="https://studyib.net/img/studyib-card-default.jpg"> <meta name="keywords" content="Differentiation, Chain rule, Product rule, Quotient rule, Graphs, Tangent, Normal, Stationary points, Kinematics, Implicit, IB, IBDP, InThinking, International Baccalaureate, Revision, Revision websites, Student Sites, Students, Learning, Guidance"> <meta itemprop="name" content="StudyIB Maths: Analysis & Approaches: HL Mixed Differentiation"> <meta itemprop="description" content="This page is ideal for practising all the skills of differentiation. You may wish to use this page in preparation for a test on this topic or for the final examinations. The quizzes on this page have been carefully created to take you through..."> <meta itemprop="image" content="https://studyib.net/img/studyib-card-default.jpg"> <meta name="twitter:card" content="summary_large_image"> <meta name="twitter:title" content="StudyIB Maths: Analysis & Approaches: HL Mixed Differentiation"> <meta name="twitter:description" content="This page is ideal for practising all the skills of differentiation. You may wish to use this page in preparation for a test on this topic or for the final examinations. The quizzes on this page have been carefully created to take you through..."> <meta name="twitter:image" content="https://studyib.net/img/studyib-card-default.jpg"> <meta name="twitter:creator" content="@inthinker"><link rel="stylesheet" href="../../../css/snippets.min.css?v=202209111300"><link rel="stylesheet" href="../../../css/article.min.css?v=202211221000"><link rel="stylesheet" type="text/css" href="../../../js/slick-carousel/slick.min.css"><link rel="stylesheet" type="text/css" href="../../../js/slick-carousel/slick-theme.min.css"><style type="text/css">.filter-sl-only { display: none; }</style><script src="../../../ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script><script src="../../../js/ifvisible.min.js"></script><script>ifvisible.setIdleDuration(300);</script><link rel="apple-touch-icon-precomposed" sizes="57x57" href="../../../img/favicon/apple-touch-icon-57x57.png"> <link rel="apple-touch-icon-precomposed" sizes="114x114" href="../../../img/favicon/apple-touch-icon-114x114.png"> <link rel="apple-touch-icon-precomposed" sizes="72x72" href="../../../img/favicon/apple-touch-icon-72x72.png"> <link rel="apple-touch-icon-precomposed" sizes="144x144" href="../../../img/favicon/apple-touch-icon-144x144.png"> <link rel="apple-touch-icon-precomposed" sizes="60x60" href="../../../img/favicon/apple-touch-icon-60x60.png"> <link rel="apple-touch-icon-precomposed" sizes="120x120" href="../../../img/favicon/apple-touch-icon-120x120.png"> <link rel="apple-touch-icon-precomposed" sizes="76x76" href="../../../img/favicon/apple-touch-icon-76x76.png"> <link rel="apple-touch-icon-precomposed" sizes="152x152" href="../../../img/favicon/apple-touch-icon-152x152.png"> <link rel="icon" type="image/png" href="../../../img/favicon/favicon-196x196.png" sizes="196x196"> <link rel="icon" type="image/png" href="../../../img/favicon/favicon-96x96.png" sizes="96x96"> <link rel="icon" type="image/png" href="../../../img/favicon/favicon-32x32.png" sizes="32x32"> <link rel="icon" type="image/png" href="../../../img/favicon/favicon-16x16.png" sizes="16x16"> <link rel="icon" type="image/png" href="../../../img/favicon/favicon-128.png" sizes="128x128"> <meta name="application-name" content="StudyIB: Your IB Learning Companion"> <meta name="msapplication-TileColor" content="#A5BED5"> <meta name="msapplication-TileImage" content="/img/favicon/mstile-144x144.png"> <meta name="msapplication-square70x70logo" content="/img/favicon/mstile-70x70.png"> <meta name="msapplication-square150x150logo" content="/img/favicon/mstile-150x150.png"> <meta name="msapplication-wide310x150logo" content="/img/favicon/mstile-310x150.png"> <meta name="msapplication-square310x310logo" content="/img/favicon/mstile-310x310.png"><script>var stdHash = "6da675cbcb3d25f050b05557349abc79", stdTicket = "082b9c9c4ae3624d";</script><script src="../../../js/user/local-stats.min.js?v=202205311700"></script><link href="../../../css/subjects-frontpage.min.css?v=202302031000" rel="stylesheet"></head><body class="public mathsanalysis"> <noscript><iframe src="https://www.googletagmanager.com/ns.html?id=GTM-PZBWZ8J" height="0" width="0" style="display:none;visibility:hidden"></iframe></noscript> <div id="top-header"> <div class="wmap general hidden-sm hidden-xs subjects"> <div class="layout-wrapper"> <div class="container-fluid"> <a href=""> <img class="header-thinker" src="../../../img/header-thinker.svg"> </a> <h1> <a href="../../../mathsanalysis.html"> <span style="font-family: 'Helvetica Narrow','Arial Narrow',Tahoma,Arial,Helvetica,sans-serif;font-stretch: condensed;letter-spacing: -1px">IBDP Maths: Analysis & Approaches</span> </a> <a href="https://inthinking.net" title="inthinking.net" class="inthinking-logo pull-right"> <img src="../../../img/header-logo.svg" style="height: 80px; width: auto;"> </a> </h1> <p class="slogan">InThinking Revision Sites for students</p> <p class="author">Website by <strong>Richard Wade</strong></p> <p class="updated">Updated 3 February 2023</p> </div> </div> </div> <nav id="public-topnav" class="navbar navbar-default"> <div class="container-fluid"> <div class="navbar-header"> <a class="navbar-brand hidden-md hidden-lg" href="../../../index.htm"> <img class="header-xs-thinker" src="../../../img/header-thinker.svg"> </a> <button type="button" class="collapsed navbar-toggle" data-toggle="collapse" data-target="#subject-navbar-collapse" aria-expanded="false"> <span class="sr-only">Toggle navigation</span> <i class="fa fa-fw fa-2x fa-user"></i> </button> <div class="brand hidden-lg hidden-md hidden-sm"> <a class="brand-xs" href="../../../mathsanalysis.html"> <strong class="title" style="white-space: nowrap;font-size: 18px;">Maths: A&A</strong> </a> </div> </div> <div class="collapse navbar-collapse" id="subject-navbar-collapse"> <ul class="nav navbar-bar navbar-userbox hidden-md hidden-lg"><li class="dropdown"><a href="#" class="dropdown-toggle" data-toggle="dropdown"><i class="fa fa-fw fa-lg fa-user-circle" style="margin-right: 5px;"></i><span style="font-size: 16px;"></span></a><ul class="dropdown-menu"><li class="dropdown-submenu"><a class="dropdown-toggle" href="#" data-toggle="dropdown"><i class="fa fa-fw fa-globe"></i> Subjects</a><ul class="dropdown-menu"><li></li><li class=""><a href="../../../biology.html">DP Biology</a></li><li class=""><a href="../../../chemistry.html">DP Chemistry</a></li><li class=""><a href="../../../englishalanglit.html">DP English A: Language & Literature</a></li><li class="active"><a href="../../../mathsanalysis.html"><i class="fa fa-caret-right"></i> DP Maths: Analysis & Approaches</a></li><li class=""><a href="../../../mathsapplications.html">DP Maths: Applications & Interpretations SL</a></li><li class=""><a href="../../../physics.html">DP Physics</a></li><li class=""><a href="../../../spanishb.html">DP Spanish B</a></li></ul></li><li class="menu-item"><a href="../../../user.html"><i class="fa fa-fw fa-dashboard"></i> Dashboard</a></li><li class="menu-item"><a href="../../../user/profile.html"><i class="fa fa-fw fa-cog"></i> My profile</a></li><li class="menu-item"><a href="../../../user/messages.html"><i class="fa fa-fw fa-envelope"></i> Messages</a></li><li class="menu-item hidden-md hidden-lg mt-xs-3"><a href="../../../index.htm?logout=1" class="btn btn-primary btn-xs-block"><i class="fa fa-fw fa-lg fa-power-off text-danger"></i>Log out</a></li></ul></li></ul> <ul class="nav navbar-bar"><li class=""><a class="home" href="../../../mathsanalysis.html"><i class="fa fa-fw fa-home"></i> Home</a></li><li class=""><a class="topics" href="#"><i class="fa fa-fw fa-th-large"></i> Topics</a></li><li class=""><a class="favorites" href="../../../mathsanalysis.html"><i class="fa fa-fw fa-star"></i> My favorites</a></li><li class=""><a class="qbank" href="../../test-your-knowledge.html"><i class="fa fa-fw fa-question"></i> Question bank</a></li><li class=""><a class="sitemap" href="../../sitemap.html"><i class="fa fa-fw fa-sitemap"></i> Sitemap</a></li><li class=""><a class="activity" href="../../../mathsanalysis.html"><i class="fa fa-fw fa-calendar-check-o"></i> My activity</a></li></ul> <ul class="nav navbar-bar navbar-right"> <li> <form class="navbar-form hidden-md hidden-lg" role="search" method="get" action="mathsanalysis/search"> <div class="input-group"> <input class="form-control nav-search" name="glob" type="search" placeholder="Search..."> <span class="input-group-btn"> <button type="submit" class="btn bg-blue"> <i class="fa fa-search"> </i> </button> </span> </div> </form> <a href="#" class="toggle-menu-search hidden-sm hidden-xs" data-toggle="dropdown"> <i class="fa fa-lg fa-search"></i> </a> </li></ul> </div> </div></nav><nav id="nav-menu-search" class="shadow-bottom hidden-sm hidden-xs" style="display: none;"> <div class="container-fluid"> <form class="" role="search" method="get" action="mathsanalysis/search"> <input class="form-control nav-search" name="glob" type="search" placeholder="Search Maths: A&A..."> <button class="btn btn-sm btn-primary" type="submit"> Search </button> </form> </div></nav><div class="modal fade" tabindex="-1" role="dialog" id="modal-mini-login"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-body"> <button aria-hidden="true" data-dismiss="modal" class="close hidden-xs login-button-close" type="button">×</button> <form method="post"> <div class="form-group"> <input name="user-email" class="form-control" type="text" placeholder="Email"> </div> <div class="form-group"> <input name="user-password" class="form-control" type="password" placeholder="Password"> <input name="user-fp" class="fp" value="d8c893da5db7501669ffeeac46273ba6" type="hidden"> </div> <div class="form-group"> <a href="../../../reset-password.html" class="reset-password" style="font-weight: normal;"> Forgot your password? </a> </div> <div class="form-group"> <button type="submit" name="submit-login" value="1" class="btn btn-primary btn-block text-center"> <i class="fa fa-user fa-fw"></i> Log in </button> </div> </form> </div> </div> </div></div></div><div class="modal fade" id="show-topics" tabindex="-1" role="dialog"><div id="dialog-topics" class="modal-dialog modal-dialog-topics"><div class="modal-content"><div class="modal-body modal-body-topics"><div id="new-frontpage-toplevels-topics" class="frontpage-box"><div class="row"><div class="col-md-4 col-sm-4"><div class="item"><a href="../3016/free-access-weekend.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Free Access Weekend</h3></div><div class="body"><div class="cropper" style="background-image: url('../../images/free.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../2696/start-here.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Start Here</h3></div><div class="body"><div class="cropper" style="background-image: url('../../images/start-here.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../2902/examination-questions.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Examination Questions</h3></div><div class="body"><div class="cropper" style="background-image: url('../../images/exam.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../2563/question-bank.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Question Bank</h3></div><div class="body"><div class="cropper" style="background-image: url('../../images/questionbank.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../537/algebra.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Algebra</h3></div><div class="body"><div class="cropper" style="background-image: url('/media/mathsanalysis/images/algebra.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../539/functions.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Functions</h3></div><div class="body"><div class="cropper" style="background-image: url('/media/mathsanalysis/images/function_s.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../540/geometry-trigonometry.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Geometry & Trigonometry</h3></div><div class="body"><div class="cropper" style="background-image: url('../../images/photo-by-a-href=httpsunsplash.com@pawel_czerwinskiutm_source=unsplash-utm_medium=referral-utm_content=creditcopytextpawel-czerwinskia-on-a-href=httpsunsplash.comutm_source=unsplash-utm_medium=referral-utm_co.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../549/stats-probability.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Stats & Probability</h3></div><div class="body"><div class="cropper" style="background-image: url('/media/mathsanalysis/images/probability.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../550/calculus.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Calculus</h3></div><div class="body"><div class="cropper" style="background-image: url('/media/mathsanalysis/images/calculus1.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../857/exam-tips.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Exam Tips</h3></div><div class="body"><div class="cropper" style="background-image: url('/media/mathsanalysis/files/exam-tips/exam-tips.jpg'); height: 10vw;"></div></div></a></div></div></div></div></div></div></div></div><div class="container-fluid shadow-ith"><div class="row frontpage"> <div class="col-md-3" id="left-column"> <p class="visible-xs-block"></p><div class="dropdown user-menu hidden-xs" style="margin-bottom: 20px;"> <button type="button" class="btn btn-default dropdown-toggle" data-toggle="dropdown" style="border-radius: 0; padding: 6px 18px;"> <i class="fa fa-fw fa-user-circle"></i> <small><i class="fa fa-caret-down"></i></small> </button> <ul class="dropdown-menu"> <li class="dropdown-submenu"><a href="#"><i class="fa fa-fw fa-globe"></i> Subjects</a><ul id="user-subjects-menu" class="dropdown-menu"><li></li><li class=""><a href="../../../biology.html">DP Biology</a></li><li class=""><a href="../../../chemistry.html">DP Chemistry</a></li><li class=""><a href="../../../englishalanglit.html">DP English A: Language & Literature</a></li><li class="active"><a href="../../../mathsanalysis.html"><i class="fa fa-caret-right"></i> DP Maths: Analysis & Approaches</a></li><li class=""><a href="../../../mathsapplications.html">DP Maths: Applications & Interpretations SL</a></li><li class=""><a href="../../../physics.html">DP Physics</a></li><li class=""><a href="../../../spanishb.html">DP Spanish B</a></li></ul></li><li class="divider"></li><li><a href="../../../user.html"><i class="fa fa-fw fa-dashboard"></i> Dashboard</a></li><li><a href="../../../user/profile.html"><i class="fa fa-fw fa-cog"></i> My profile</a></li><li><a href="../../../user/messages.html"><i class="fa fa-fw fa-envelope"></i> Messages</a></li> <li class="divider"></li> <li> <a href="../../../index.htm?logout=1"> <i class="fa fa-fw fa-power-off"></i> Log out </a> </li> </ul></div> <div id="side-nav"><h4 class="side-nav-title dropdown"><a href="../550/calculus.html" class="dropdown-toggle toplevel-dropdown" data-toggle="dropdown"><i class="fa fa-ellipsis-v"></i></a> <a href="../550/calculus.html">Calculus</a><ul class="dropdown-menu"><li><a href="../../../mathsanalysis.html" style="padding-left: 5px; border-bottom: solid 1px #ddd"><i class="fa fa-fw fa-home"></i> Home</a></li><li><a href="../3016/free-access-weekend.html"><i class="fa fa-fw fa-caret-right"></i> Free Access Weekend</a></li><li><a href="../2696/start-here.html"><i class="fa fa-fw fa-caret-right"></i> Start Here</a></li><li><a href="../2902/examination-questions.html"><i class="fa fa-fw fa-caret-right"></i> Examination Questions</a></li><li><a href="../2563/question-bank.html"><i class="fa fa-fw fa-caret-right"></i> Question Bank</a></li><li><a href="../537/algebra.html"><i class="fa fa-fw fa-caret-right"></i> Algebra</a></li><li><a href="../539/functions.html"><i class="fa fa-fw fa-caret-right"></i> Functions</a></li><li><a href="../540/geometry-trigonometry.html"><i class="fa fa-fw fa-caret-right"></i> Geometry & Trigonometry</a></li><li><a href="../549/stats-probability.html"><i class="fa fa-fw fa-caret-right"></i> Stats & Probability</a></li><li><a href="../857/exam-tips.html"><i class="fa fa-fw fa-caret-right"></i> Exam Tips</a></li></ul><div class="pull-right"><a class="sidenav-expand" title="Expand all" href="#"><i class="fa fa-plus-circle"></i></a> <a class="sidenav-compress" title="Compress all" href="#"><i class="fa fa-minus-circle"></i></a></div></h4><ul class="side-nav level-0"><li class="expanded parent selected"><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="hl-mixed-differentiation.html">HL Mixed Differentiation</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../1165/mixed-integration.html">Mixed Integration</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../657/introducing-derivatives.html">Introducing Derivatives</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../658/graphs-and-derivatives.html">Graphs and Derivatives</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../742/chain-rule.html">Chain Rule</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../743/product-and-quotient-rule.html">Product and Quotient Rule</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../744/equation-of-tangent-and-normal.html">Equation of Tangent and Normal</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../695/optimisation.html">Optimisation</a></label></li><li class=""><label style="padding-left: 0px"><a class="expander" href="#" style="font-size: .9em"><i class="fa fa-fw fa-caret-right"></i></a><a href="../741/implicit-differentiation.html">Implicit Differentiation</a></label><ul class="side-nav level-1"><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../565/graphs-of-implicit-equations.html">Graphs of Implicit Equations</a></label></li></ul></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../2872/related-rates-of-change.html">Related Rates of Change</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../2829/lhopitals-rule.html">L'Hôpital's Rule</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../586/definite-integration.html">Definite Integration</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../730/area-between-graphs-hl.html">Area between Graphs HL</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../644/volume-of-revolution.html">Volume of Revolution</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../634/integration-by-substitution-hl.html">Integration by Substitution HL</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../635/integration-by-parts.html">Integration by Parts</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../2107/differential-equations-separable-variables.html">Differential Equations - Separable Variables</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../2109/differential-equations-integrating-factor.html">Differential Equations - Integrating Factor</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../2108/differential-equations-homogeneous.html">Differential Equations - Homogeneous</a></label></li></ul></div> <div class="hidden-xs hidden-sm"> <button class="btn btn-default btn-block text-xs-center" data-toggle="modal" data-target="#modal-feedback" style="margin-bottom: 10px"><i class="fa fa-send"></i> Feedback</button> </div> </div> <div class="col-md-9" id="main-column"> <h1 class="page_title"> HL Mixed Differentiation <a href="#" class="mark-page-favorite pull-right" data-pid="2405" title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsanalysis.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../550/calculus.html">Calculus</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">HL Mixed Differentiation</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 30 minutes"><i class="fa fa-clock-o"></i> 30'</span> </ol> <article id="main-article"> <p><img alt="" src="../../files/differentiation/mixed-differentiation/calculus-1.jpg" style="float: left; width: 150px; height: 100px;"></p> <p>This page is ideal for practising all the skills of differentiation. You may wish to use this page in preparation for a test on this topic or for the final examinations. The quizzes on this page have been carefully created to take you through all the skills that you need. If you want a more in depth look, then you should go to the individual pages on these topics.</p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <p>On this page, you can revise</p> <ul> <li>differentiating x<sup>n</sup></li> <li>differentiating <em>f(x) + g(x)</em></li> <li>differentiating <em>sinx</em>, <em>cosx</em>, <em>e<sup>x</sup></em> and <em>lnx</em></li> <li>the chain rule</li> <li>the product rule</li> <li>the quotient rule</li> <li>related rates of change</li> <li>differentiating using a graphical display calculator</li> <li>graphs and the gradient function</li> <li>stationary points</li> <li>equations of tangents and normals</li> <li>kinematics</li> </ul> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <p>Here's a quiz that practises differentiating <em><strong>x<sup>n</sup></strong></em><strong><em> </em></strong></p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#8e362e53"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="8e362e53"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Differentiating x to the n (2nd) <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-596-2405" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(2x^3)=ax^b\)</span>, find <strong><em>a</em></strong> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="6"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p>Use the power rule <span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(ax^n)=nax^{n-1}\)</span></p><p><span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(2x^3)=6x^2\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(\large y = 3x^5\)</span>, then <span class="math-tex">\(\large\frac{\mathrm{d}y}{\mathrm{d}x}=ax^b\)</span></p><p>Find <strong><em>a</em></strong> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="15"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p></div><div class="q-explanation"><p>Use the power rule <span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(ax^n)=nax^{n-1}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(f(x)=2x^{-1}\)</span>, then <span class="math-tex">\(f'(x)=ax^{b}\)</span></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></p></div><div class="q-explanation"><p>Use the power rule <span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(ax^n)=nax^{n-1}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Find <span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(5x)\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{5x^2}{2}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> 5</label></p><p><label class="radio"><input type="radio"> 5x</label></p><p><label class="radio"><input type="radio"> 0</label></p></div><div class="q-explanation"><p>What is the gradient of the straight line y = 5x ?</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(\large A=4\pi r^2\)</span>, find <span class="math-tex">\(\large\frac{\mathrm{d}A}{\mathrm{d}r}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 8r</label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large 8 \pi r\)</span></label></p><p><label class="radio"><input type="radio"> 2r</label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large 4 \pi \)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(4\pi \)</span> is a constant</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(\large f(x) = 10\)</span> , what is <span class="math-tex">\(\large f'(x)\)</span> ?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 10x</label></p><p><label class="radio"><input class="c" type="radio"> 0</label></p><p><label class="radio"><input type="radio"> 10</label></p><p><label class="radio"><input type="radio"> not possible</label></p></div><div class="q-explanation"><p>The derivative of a constant is zero</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(\large V=\frac{4}{3}\pi r^3\)</span> , what is <span class="math-tex">\(\large\frac{\mathrm{d}V}{\mathrm{d}r}\)</span>?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{4}{3}\pi r^2\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{4}{9}\pi r^2\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large 4\pi r^2\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{4}{3}\pi \)</span></label></p></div><div class="q-explanation"><p>Use the power rule</p><p><span class="math-tex">\(\large \frac{4}{3}\pi\)</span> is a constant</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x^2})=ax^b\)</span></p><p>Work out <strong><em>a</em></strong> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><em><strong>a</strong></em> = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="-3"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x^2})=\large\frac{\mathrm{d}}{\mathrm{d}x}(x^{-2})=-2x^{-3}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2}{x^3})=ax^b\)</span></p><p>Work out <strong><em>a</em></strong> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="-6"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="-4"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2}{x^3})=\large\frac{\mathrm{d}}{\mathrm{d}x}(2x^{-3})=-6x^{-4}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>What is <span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3}{\sqrt{x}})\)</span>?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large\frac{3}{2}x^{- \frac {1}{2}}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large -\frac{6}{\sqrt{x^3}}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large-\frac{3}{2}\sqrt{x}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large -\frac{3}{2}x^{-\frac{3}{2}}\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3}{\sqrt{x}})=\frac{\mathrm{d}}{\mathrm{d}x}(3x^{-\frac{1}{2}})=-\frac{3}{2}x^{-\frac{3}{2}}=-\frac{3}{2\sqrt{x^3}}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> <hr class="hidden-separator"> <p>Here's a quiz that practises differentiating functions in the form <em><strong>f(x) + g(x)</strong></em></p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#dbdcda37"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="dbdcda37"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Differentiating f(x)+g(x) <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-597-2405" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p><span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(x^3+x^2)=ax^2+2x^b\)</span></p><p>Work out <strong><em>a</em></strong> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"><p>Use the power rule <span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(ax^n)=nax^{n-1}\)</span></p><p><span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(x^3+x^2)=3x^2+2x\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(2x^4-3x^2)=8x^a+bx\)</span></p><p>Work out <strong><em>a</em></strong> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="-6"> <span class="review"></span></p></div><div class="q-explanation"><p>Use the power rule <span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(ax^n)=nax^{n-1}\)</span></p><p><span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(2x^4-3x^2)=8x^3-6x\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out <span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(\sqrt{x}-\frac{1}{x})\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large\frac{1}{2}x^{-\frac{1}{2}}-x^{-2}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large \frac{1}{2\sqrt{x}}+\frac{1}{x^2}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large\frac{2}{\sqrt{x}}+x^0\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large2x^{-\frac{1}{2}}+x^{2}\)</span></label></p></div><div class="q-explanation"><p>Use the power rule <span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(ax^n)=nax^{n-1}\)</span></p><p><span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(\sqrt{x}-\frac{1}{x})=\large\frac{\mathrm{d}}{\mathrm{d}x}(x^\frac{1}{2}-x^{-1})=\frac{1}{2}x^{-\frac{1}{2}}+x^{-2}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(\large y=x(x^2-2)\)</span> , work out <span class="math-tex">\(\large\frac{\mathrm{d}y}{\mathrm{d}x}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large x^3-2x\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large3x^2-2\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large (x^2-2)+x(2x)\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large 2x\)</span></label></p></div><div class="q-explanation"><p>There is no need to use the product rule for differentiation. Expand the brackets first</p><p><span class="math-tex">\(\large y=x(x^2-2)=x^3-2x\)</span></p><p><span class="math-tex">\(\large\frac{\mathrm{d}y}{\mathrm{d}x}=3x^2-2\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(\large f(x)=(x-1)(x+2)\)</span> , work out <span class="math-tex">\(f'(x)\)</span></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large 2x+1\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large 2x-1\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large 1(x+2)-1(x-1)\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large x^2+x-2\)</span></label></p></div><div class="q-explanation"><p>There is no need to use the product rule for differentiation. Expand the brackets first</p><p><span class="math-tex">\(\large f(x)=(x-1)(x+2)=x^2+2x-x-2=x^2+x-2\)</span></p><p><span class="math-tex">\(\large f'(x)=2x+1\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out <span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^3-1}{x})\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{2x^3-1}{x^2}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large x^2-x^{-1}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large 2x-\frac{1}{x^2}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large \frac{2x^3+1}{x^2}\)</span></label></p></div><div class="q-explanation"><p>There is no need to use the quotient rule. Simple before differentiating</p><p><span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^3-1}{x})=\large\frac{\mathrm{d}}{\mathrm{d}x}(x^2- \frac{1}{x})=\large\frac{\mathrm{d}}{\mathrm{d}x}(x^2-x^{-1})\\
=2x+x^{-2}=2x+\frac{1}{x}\\
=\frac{2x^3+1}{x^2}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> If <span class="math-tex">\(\large y = 5 - 3x^2\)</span> , work out <span class="math-tex">\(\large\frac{\mathrm{d}y}{\mathrm{d}x}\)</span> when <span class="math-tex">\(\large x = \frac{1}{2}\)</span></p></div><div class="q-answer"><p><span class="math-tex">\(\large\frac{\mathrm{d}y}{\mathrm{d}x}\)</span>= <input type="text" style="height: auto;" data-c="-3"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large\frac{\mathrm{d}y}{\mathrm{d}x}=-6x\)</span></p><p>When <span class="math-tex">\(\large x = \frac{1}{2}\)</span>, <span class="math-tex">\(\large\frac{\mathrm{d}y}{\mathrm{d}x}=-6\times \frac{1}{2}=-3\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> If <span class="math-tex">\(\large f(x)=x^2(2x-1)\)</span> , find <span class="math-tex">\(\large f'(-1)\)</span></p></div><div class="q-answer"><p><span class="math-tex">\(\large f'(-1)\)</span> = <input type="text" style="height: auto;" data-c="8"> <span class="review"></span></p></div><div class="q-explanation"><p>Expand the brackets before differentiating</p><p><span class="math-tex">\(\large f(x)=x^2(2x-1)=2x^3-x^2\)</span></p><p><span class="math-tex">\(\large f'(x)=6x^2-2x\)</span></p><p><span class="math-tex">\(\large f'(-1)=6(-1)^2-2(-1)=6+2=8\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> If <span class="math-tex">\(\large y=\frac{x-4}{x^2}\)</span> , work out <span class="math-tex">\(\large\frac{\mathrm{d}y}{\mathrm{d}x}\)</span>when x = 1</p></div><div class="q-answer"><p><span class="math-tex">\(\large\frac{\mathrm{d}y}{\mathrm{d}x}\)</span> = <input type="text" style="height: auto;" data-c="7"> <span class="review"></span></p></div><div class="q-explanation"><p>There is no need to use the quotient rule for differentiation. Simplify before differentiating</p><p><span class="math-tex">\(\large y=\frac{x-4}{x^2}=\frac{1}{x}-\frac{4}{x^2}=x^{-1}-4x^{-2}\)</span></p><p><span class="math-tex">\(\large\frac{\mathrm{d}y}{\mathrm{d}x}=-x^{-2}+8x^{-3}\)</span></p><p><span class="math-tex">\(\large\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{1}{x^2}+\frac{8}{x^3}\\
\)</span></p><p>When x = 1 , <span class="math-tex">\(\large\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{1}{1}+\frac{8}{1}=7\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> If <span class="math-tex">\(\large f(x)=2x^2-\sqrt{x}\)</span> , find <span class="math-tex">\(\large f'(0.25)\)</span></p></div><div class="q-answer"><p><span class="math-tex">\(\large f'(0.25)\)</span> = <input type="text" style="height: auto;" data-c="0"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large f(x)=2x^2-x^{\frac{1}{2}}\)</span></p><p><span class="math-tex">\(\large f'(x)=4x-\frac{1}{2}x^{-\frac{1}{2}}=4x-\frac{1}{2\sqrt{x}}\)</span></p><p><span class="math-tex">\(\large f'(0.25)=4(0.25)-\frac{1}{2\sqrt{0.25}}=1-\frac{1}{2\times 0.5}=1-1=0\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> <hr class="hidden-separator"> <p>Here's a quiz that practises differentiating functions in the form <em><strong>sinx</strong></em>, <em><strong>cosx</strong></em>, <em><strong>e<sup>x</sup></strong></em> and <em><strong>lnx</strong></em></p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#67f5f316"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="67f5f316"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Differentiating sinx, cosx, ex and lnx <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-598-2405" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(\large y = \sin x - \cos x\)</span> , find <span class="math-tex">\(\large\frac{\mathrm{d}y}{\mathrm{d}x}\)</span></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large \cos x+\sin x \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large- \cos x-\sin x \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\( \large \cos x-\sin x \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large\sin x- \cos x \)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(\sin x)=\cos x\\ \large\frac{\mathrm{d}}{\mathrm{d}x}(\cos x)=-\sin x\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(\large f(x) = 3\sin x +a \cos x\)</span> , and <span class="math-tex">\(\large f'(x)=b\cos x-2 \sin x\)</span></p><p>Find <strong><em>a</em></strong> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(3\sin x)=3\cos x\\ \large\frac{\mathrm{d}}{\mathrm{d}x}(2\cos x)=-2\sin x\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(a\sin x+b \cos x)=5 \sin x-3 \cos x\)</span></p><p>Find <strong><em>a</em></strong> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="-3"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="-5"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(-5\cos x)=5\sin x\\ \large\frac{\mathrm{d}}{\mathrm{d}x}(-3\sin x)=-3\cos x\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> If <span class="math-tex">\(\large f(x) = 2\sin x \)</span> , find <span class="math-tex">\(\large f'(\frac{\pi}{3})\)</span></p></div><div class="q-answer"><p><span class="math-tex">\(\large f'(\frac{\pi}{3})\)</span> = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large f'(x) = 2\cos x \\
\large f'(\frac{\pi}{3}) = 2\cos (\frac{\pi}{3}) =2\times\frac{1}{2}=1\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> If <span class="math-tex">\(\large f(x) = 2\cos x \)</span> , and <span class="math-tex">\(\large f'(\frac{\pi}{a})=-\sqrt{2}\)</span></p><p>find <strong><em>a</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large f'(x) = -2\sin x \\
\large f'(\frac{\pi}{a}) = -2\sin (\frac{\pi}{a})=-\sqrt{2}\)</span></p><p><span class="math-tex">\(\large\sin (\frac{\pi}{a})=\frac{\sqrt{2}}{2}\)</span></p><p><span class="math-tex">\(\large\sin (\frac{\pi}{4})=\frac{\sqrt{2}}{2}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out <span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(x^2+e^x)\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large x^2+e^x\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large2x+e^x\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large2x+xe^{x-1}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large2x+e^{x-1}\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(e^x)=e^x\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(\large f(x) = \frac{\ln x}{2}\)</span> , work out <span class="math-tex">\(\large f'(x)\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{1}{\frac{x}{2}}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large \frac{1}{2x}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{2}{x}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{e^x}{2}\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(\ln x)=\frac{1}{x}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out <span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(\ln 2x)\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large2x\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large\frac{2}{x}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large\frac{1}{x}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large\frac{1}{2x}\)</span></label></p></div><div class="q-explanation"><p>Using the laws of logarithms, we know that</p><p><span class="math-tex">\(\large\ln 2x=\ln 2+\ln x
\)</span></p><p><span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(\ln 2+\ln x)=0+\frac{1}{x}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> <span class="math-tex">\(\large y=\frac{2e^x+1}{3}\)</span></p><p>Find <span class="math-tex">\(\large\frac{\mathrm{d}y}{\mathrm{d}x}\)</span> when x = 0</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 2</label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{2}{3}e\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large \frac{2}{3}\)</span></label></p><p><label class="radio"><input type="radio"> 1</label></p></div><div class="q-explanation"><p><span class="math-tex">\(\large y=\frac{2e^x+1}{3}=\frac{2}{3}e^x+\frac{1}{3}\)</span></p><p><span class="math-tex">\(\large\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{2}{3}e^x\)</span></p><p>when x = 0 , <span class="math-tex">\(\large\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{2}{3}e^0=\frac{2}{3} \times 1=\frac{2}{3}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> <span class="math-tex">\(\large f(x) = 3e^x-2 \sin x\)</span> , work out <span class="math-tex">\(\large f'(0)\)</span></p></div><div class="q-answer"><p><span class="math-tex">\(\large f'(0)\)</span> = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large f'(x) = 3e^x-2 \cos x\)</span></p><p><span class="math-tex">\(\large f'(0) = 3e^0-2 \cos 0=3\times 1-2\times 1=1\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> <hr class="hidden-separator"> <p>Here's a quiz that mainly practises the <em><strong>Chain Rule</strong></em></p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#efeaf540"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="efeaf540"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Mixed SL Differentiation 1 <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-381-2405" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Find the derivative of (2x - 5)<sup>4</sup></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> (2x - 5)<sup>3</sup></label></p><p><label class="radio"><input class="c" type="radio"> 8(2x - 5)<sup>3</sup></label></p><p><label class="radio"><input type="radio"> 4(2x - 5)<sup>3</sup></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{(2x-5)^{3}}{8}\)</span></label></p></div><div class="q-explanation"><p>We can use the chain rule</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>y = u<sup>4</sup></td><td>u = 2x - 5</td></tr><tr><td><span class="math-tex">\(\frac { dy }{ du } \)</span> = 4u<sup>3</sup></td><td><span class="math-tex">\(\frac { du }{ dx } \)</span> = 2</td></tr></tbody></table><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { dy }{ du } \times \frac { du }{ dx } \)</span></p><p>= 8u<sup>3</sup></p><p>=8(2x - 5)<sup>3</sup></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the derivative of e<sup>5x</sup></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> e<sup>5x</sup></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{e^{5x}}{5}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> 5e<sup>5x</sup></label></p><p><label class="radio"><input type="radio"> 5e<sup>4x</sup></label></p></div><div class="q-explanation"><p>You can use the chain rule, but you should try to remember that <span class="math-tex">\(\frac{d}{dx}(e^{ax})=ae^{ax}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the gradient function of f(x) = ln3x</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(f'(x)=\frac{3}{x}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(f'(x)=\frac{1}{3x}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(f'(x)=\frac{1}{x}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(f'(x)=3ln2x\)</span></label></p></div><div class="q-explanation"><p>There are 2 ways of thinking about this type of derivative.</p><ul><li>You can use the chain rule <span class="math-tex">\(f'(x)=\frac{3}{3x}=\frac{1}{x}\)</span></li></ul><p>or</p><ul><li>You can use the properties of logarithms</li></ul><p>f(x) = ln3x = ln3 + lnx</p><p><span class="math-tex">\(f'(x)=0+\frac{1}{x}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Find <span class="math-tex">\(\frac { dy }{ dx } \)</span> if <span class="math-tex">\(y =\frac{1}{e^{x^{3}}}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { 1}{ e^{x^{3}}}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { 3x^{2} }{ e^{x^{3}}}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(-\frac { 3x^{2} }{ e^{x^{3}}}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { 1 }{ e^{3x^{2}}}\)</span></label></p></div><div class="q-explanation"><p>Notice that we can re-write the question to make it easier</p><p><span class="math-tex">\(y =\frac{1}{e^{x^{3}}}=e^{-x^{3}}\)</span></p><p>We can use the chain rule</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>y = e<sup>u</sup></td><td>u = <span class="math-tex">\(-x^{3}\)</span></td></tr><tr><td><span class="math-tex">\(\frac { dy }{ du } \)</span> = e<sup>u</sup></td><td><span class="math-tex">\(\frac { du }{ dx } \)</span> = -3x²</td></tr></tbody></table><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { dy }{ du } \times \frac { du }{ dx } \)</span></p><p>= e<sup>u</sup>(-3x²)</p><p>=<span class="math-tex">\(-3x^{2}e^{-x^{3}}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the derivative of <span class="math-tex">\( {1\over 4x}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\( {1\over 4}lnx\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(- {1\over 4x²}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\( {1\over 4}\)</span></label></p><p><label class="radio"><input type="radio"> ln4x</label></p></div><div class="q-explanation"><p>It helps if you think of the question like this</p><p><span class="math-tex">\( {1\over 4x}= {1\over 4} x^{-1}\)</span></p><p>Some people get integration and differentiation mixed up for logarithm functions</p><p>Note that <span class="math-tex">\(\int { \frac { 1 }{ 4x } dx= } \frac { 1 }{ 4 } \int { \frac { 1 }{ x } dx= } \frac { 1 }{ 4 } lnx
\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Find <span class="math-tex">\(\frac { d }{ dx } (lnx)^{3}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 3(lnx)²</label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(({1 \over x})^{3}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac{3lnx^{2}}{x}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\({3\over x}\)</span></label></p></div><div class="q-explanation"><p>Use the chain for this, or use the quick rule</p><p><span class="math-tex">\(\frac { d }{ dx } [f(x)]^{ n }=nf'(x)[f(x)]^{ n-1 }\)</span></p><p><span class="math-tex">\(\frac { d }{ dx }(lnx)^{3}=3{1\over x}(lnx)^{2}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Find <span class="math-tex">\(\frac { d }{ dx } (lnx^{3})\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\({1\over x^{3}}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\({3\over x}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{3lnx^{2}}{x}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\({1 \over x^{3}}\)</span></label></p></div><div class="q-explanation"><p>Use laws of logarithms to transform this question into something easier</p><p><span class="math-tex">\(lnx^{3}=3lnx\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Differentiate sin2x</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> 2cos2x</label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(-{cos2x \over 2}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\({cos2x \over 2}\)</span></label></p><p><label class="radio"><input type="radio"> -2cos2x</label></p></div><div class="q-explanation"><p>You should try to remember that <span class="math-tex">\(\frac { d }{ dx } (sinax)=acosax\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Find <span class="math-tex">\(\frac { d }{ dx } (sinx^{2})\)</span></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(2x(cosx^{2})\)</span></label></p><p><label class="radio"><input type="radio"> 2xcos2x</label></p><p><label class="radio"><input type="radio"> 2xcosx</label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\({cosx^{2} \over 2x}\)</span></label></p></div><div class="q-explanation"><p>We can use the chain rule</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>y = sinu</td><td>u = x²</td></tr><tr><td><span class="math-tex">\(\frac { dy }{ du } \)</span> = cosu</td><td><span class="math-tex">\(\frac { du }{ dx } \)</span> = 2x</td></tr></tbody></table><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { dy }{ du } \times \frac { du }{ dx } \)</span></p><p>= cosu(2x)</p><p>= <span class="math-tex">\(2x(cosx^{2})\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Differentiate <span class="math-tex">\(\sqrt{sinx}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\( {1 \over 2}cosx \sqrt{sinx}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac{cosx}{2\sqrt{sinx}}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(cosx(sinx)^{-\frac{1}{2}}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\( {1\over 2}(sinx)^{-\frac{1}{2}}\)</span></label></p></div><div class="q-explanation"><p>We can use the chain rule</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>y = <span class="math-tex">\(u^{\frac{1}{2}}\)</span></td><td>u = sinx</td></tr><tr><td><span class="math-tex">\(\frac { dy }{ du } \)</span> = <span class="math-tex">\(\frac{1}{2}u^{-\frac{1}{2}}\)</span></td><td><span class="math-tex">\(\frac { du }{ dx } \)</span> = cosx</td></tr></tbody></table><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { dy }{ du } \times \frac { du }{ dx } \)</span></p><p>= <span class="math-tex">\(\frac{1}{2}u^{-\frac{1}{2}}cosx\)</span></p><p>= <span class="math-tex">\(\frac{1}{2}(sinx)^{-\frac{1}{2}}cosx\)</span></p><p>=<span class="math-tex">\(\frac{cosx}{2\sqrt{sinx}}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> <hr class="hidden-separator"> <p>Here's a quiz that mainly practises the <em><strong>Chain Rule and the Product and Quotient Rule</strong></em></p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#18723f71"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="18723f71"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Mixed SL Differentiation 2 <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-382-2405" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Find <span class="math-tex">\(\frac { d }{ dx } ({ e }^{ \pi -x })\)</span></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(-{ e }^{ \pi -x }\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\((\pi-1){ e }^{ \pi -x }\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\pi{ e }^{ \pi -x }\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\({ e }^{ \pi -1 }\)</span></label></p></div><div class="q-explanation"><p>We can use the chain rule</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>y = e<sup>u</sup></td><td>u = <span class="math-tex">\(\pi-x\)</span></td></tr><tr><td><span class="math-tex">\(\frac { dy }{ du } \)</span> = e<sup>u</sup></td><td><span class="math-tex">\(\frac { du }{ dx } \)</span> = -1</td></tr></tbody></table><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { dy }{ du } \times \frac { du }{ dx } \)</span></p><p>= e<sup>u</sup>(-1)</p><p>=<span class="math-tex">\(-{ e }^{ \pi -x }\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out <span class="math-tex">\(\frac { d }{ dx } [cos(2x-\frac { \pi }{ 4 } )]\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { \pi }{ 4 }sin(2x-\frac { \pi }{ 4 } )\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\((-2+\frac { \pi }{ 4 })sin(2x-\frac { \pi }{ 4 } )\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(2sin(2x-\frac { \pi }{ 4 } )\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(-2sin(2x-\frac { \pi }{ 4 } )\)</span></label></p></div><div class="q-explanation">We can use the chain rule<table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>y = cosu</td><td>u = <span class="math-tex">\(2x-\frac { \pi }{ 4 } \)</span></td></tr><tr><td><span class="math-tex">\(\frac { dy }{ du } \)</span> = -sinu</td><td><span class="math-tex">\(\frac { du }{ dx } \)</span>=2</td></tr></tbody></table><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { dy }{ du } \times \frac { du }{ dx } \)</span></p><p>=-sinu (2)</p><p>=<span class="math-tex">\(-2sin(2x-\frac { \pi }{ 4 } )\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><span class="math-tex">\(f(x) = tan(\pi -2x)\)</span></p><p>Work out <span class="math-tex">\(f'(x)\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(2{ sec }^{ 2 }(\pi -2x)\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\((\pi-2x){ sec }^{ 2 }(\pi -2x)\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\((\pi-2){ sec }^{ 2 }(\pi -2x)\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(-2{ sec }^{ 2 }(\pi -2x)\)</span></label></p></div><div class="q-explanation"><p>We can use the chain rule</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>y = tanu</td><td>u = <span class="math-tex">\( { \pi }-2x\)</span></td></tr><tr><td><span class="math-tex">\(\frac { dy }{ du } \)</span> = sec²u</td><td><span class="math-tex">\(\frac { du }{ dx } \)</span>=-2</td></tr></tbody></table><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { dy }{ du } \times \frac { du }{ dx } \)</span></p><p>=sec²u (-2)</p><p>=<span class="math-tex">\(-2{ sec }^{ 2 }(\pi -2x)\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Differentiate <span class="math-tex">\(3x(2x-1)^5\)</span></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(3(2x-1)^{4}(12x-1)\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(30(2x-1)^{4}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(3(2x-1)^{5}+3x(12x-1)^4\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(3(2x-1)^{5}+15x(12x-1)^4\)</span></label></p></div><div class="q-explanation"><p>We need to use the product rule</p><p><span class="math-tex">\(y=uv\\ \frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}\)</span></p><hr class="hidden-separator"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>u = 3x</td><td>v = <span class="math-tex">\((2x-1)^5\)</span></td></tr><tr><td><span class="math-tex">\(\frac { du }{ dx } \)</span> = 3</td><td><span class="math-tex">\(\frac { dv }{ dx } =10(2x-1)^4\)</span></td></tr></tbody></table><p><span class="math-tex">\( \frac{dy}{dx}=3x\times10(2x-1)^{4} \ +\ (2x-1)^{5}\times 3\)</span></p><p>We can factorise this result. There is a common factor of <span class="math-tex">\(3(2x-1)^4\)</span></p><p><span class="math-tex">\( \frac{dy}{dx}=3(2x-1)^{4}[10x+(2x-1)]\)</span></p><p><span class="math-tex">\( \frac{dy}{dx}=3(2x-1)^{4}(12x-1)\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Differentiate sinxcos2x</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> -2cosxsin2x</label></p><p><label class="radio"><input type="radio"> 2sinxsin2x + cosxcos2x</label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(cosxcos2x-\frac{sinxsin2x}{2}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> cosxcos2x - 2sinxsin2x</label></p></div><div class="q-explanation"><p>We need to use the product rule</p><p><span class="math-tex">\(y=uv\\ \frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}\)</span></p><hr class="hidden-separator"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>u = sinx</td><td>v = cos2x</td></tr><tr><td><span class="math-tex">\(\frac { du }{ dx } \)</span> = cosx</td><td><span class="math-tex">\(\frac { dv }{ dx } =-2sin2x\)</span></td></tr></tbody></table><p><span class="math-tex">\( \frac{dy}{dx}=sinx\times(-2sin2x) \ +cosx\times cos2x\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Differentiate <span class="math-tex">\(x^{2}lnx\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 2xlnx + x²e<sup>x</sup></label></p><p><label class="radio"><input type="radio"> 2xlnx</label></p><p><label class="radio"><input class="c" type="radio"> x + 2xlnx</label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(2xlnx+{ 1 \over x}\)</span></label></p></div><div class="q-explanation"><p>We need to use the product rule</p><p><span class="math-tex">\(y=uv\\ \frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}\)</span></p><hr class="hidden-separator"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>u = x²</td><td>v = lnx</td></tr><tr><td><span class="math-tex">\(\frac { du }{ dx } \)</span> = 2x</td><td><span class="math-tex">\(\frac { dv }{ dx } ={1 \over x}\)</span></td></tr></tbody></table><p><span class="math-tex">\( \frac{dy}{dx}=x^2 \times{1 \over x} \ +lnx\times 2x\)</span></p><p>= x + 2xlnx</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out <span class="math-tex">\(\frac { d }{ dx } (2x\sqrt { 1-{ x }^{ 2 } } )\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(2\sqrt{1-x^2} \ +\ \frac{2x}{\sqrt{1-x^2}} \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(2x\sqrt{1-x^2} \ -\ \frac{2x}{\sqrt{1-x^2}} \)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac{2-4x^2}{\sqrt{1-x^2}} \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(2\sqrt{1-x^2} \ -\ \frac{4x}{\sqrt{1-x^2}} \)</span></label></p></div><div class="q-explanation"><p>We need to use the product rule</p><p><span class="math-tex">\(y=uv\\ \frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}\)</span></p><hr class="hidden-separator"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>u = 2x</td><td>v = <span class="math-tex">\(\sqrt{1-x^2}=(1-x^2)^{1 \over 2}\)</span></td></tr><tr><td><span class="math-tex">\(\frac { du }{ dx } \)</span> = 2</td><td><span class="math-tex">\(\frac { dv }{ dx } ={1 \over 2}(-2x)(1-x^2)^{-\frac{1}{2}}=\frac{-x}{\sqrt{1-x^2}}\)</span></td></tr></tbody></table><p><span class="math-tex">\( \frac{dy}{dx}=2x \times\frac{-x}{\sqrt{1-x^2}} \ +\sqrt{1-x^2}\times 2\)</span></p><p><span class="math-tex">\(=\frac{-2x^2}{\sqrt{1-x^2}} \ +2\sqrt{1-x^2}\)</span></p><p><span class="math-tex">\(=\frac{-2x^2}{\sqrt{1-x^2}} \ + \ \frac{2(1-x^2)}{\sqrt{1-x^2}}\)</span></p><p><span class="math-tex">\(=\frac{2-4x^2}{\sqrt{1-x^2}} \)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Differentiate <span class="math-tex">\(\frac{e^x}{sinx}\)</span></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac { e^x(sinx - cosx) }{( sinx)^{ 2 }} \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { e^xcosx -e^x sinx}{( sinx)^{ 2 }} \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{e^x(sinx-cosx)}{sinx}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{e^x}{cosx}\)</span></label></p></div><div class="q-explanation"><p>We need to use the quotient rule</p><p><span class="math-tex">\(y=\frac { u }{ v } \\ \frac { dy }{ dx } =\frac { v\cdot \frac { du }{ dx } -u\cdot \frac { dv }{ dx } }{ v^{ 2 } } \)</span></p><hr class="hidden-separator"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><span class="math-tex">\(u = e^x\)</span></td><td>v = sinx</td></tr><tr><td><span class="math-tex">\(\frac{du}{dx}=e^x\)</span></td><td><span class="math-tex">\(\frac{dv}{dx}=cosx\)</span></td></tr></tbody></table><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { sinx\cdot e^x -e^x\cdot cosx }{( sinx)^{ 2 }} \)</span></p><p><span class="math-tex">\(=\frac { e^x(sinx - cosx) }{( sinx)^{ 2 }} \)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out <span class="math-tex">\(\frac{d}{dx}(\frac{e^{3x}}{cos2x})\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { e^{3x}(cos2x -sin2x) }{ (cos2x)^{ 2 } }\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac { 3e^{3x}cos2x +2e^{3x}sin2x }{ (cos2x)^{ 2 } }\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { e^{3x}(3cos2x -2sin2x) }{ (cos2x)^{ 2 } }\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { e^{3x}(cos2x +sin2x) }{ (cos2x)^{ 2 } }\)</span></label></p></div><div class="q-explanation"><p>We need to use the quotient rule</p><p><span class="math-tex">\(y=\frac { u }{ v } \\ \frac { dy }{ dx } =\frac { v\cdot \frac { du }{ dx } -u\cdot \frac { dv }{ dx } }{ v^{ 2 } } \)</span></p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><span class="math-tex">\(u=e^{3x}\)</span></td><td>v = cos2x</td></tr><tr><td><span class="math-tex">\(\frac{du}{dx}=3e^{3x}\)</span></td><td><span class="math-tex">\(\frac{dv}{dx}=-2sin2x\)</span></td></tr></tbody></table><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { cos2x\cdot 3e^{3x} -e^{3x}\cdot (-2sin2x) }{ (cos2x)^{ 2 } }\)</span></p><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { 3e^{3x}cos2x +2e^{3x}sin2x }{ (cos2x)^{ 2 } }\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Differentiate <span class="math-tex">\(\frac{x^2+1}{3x-2}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { -3x^2+4x +1}{ (3x-2)^{ 2 } } \)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac { 3x^2-4x -3}{ (3x-2)^{ 2 } } \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { 3x^2-4x +3}{ (3x-2)^{ 2 } } \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { -3x^2-4x +1}{ (3x-2)^{ 2 } } \)</span></label></p></div><div class="q-explanation"><p>We need to use the quotient rule</p><p><span class="math-tex">\(y=\frac { u }{ v } \\ \frac { dy }{ dx } =\frac { v\cdot \frac { du }{ dx } -u\cdot \frac { dv }{ dx } }{ v^{ 2 } } \)</span></p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>u = x² + 1</td><td>v = 3x - 2</td></tr><tr><td><span class="math-tex">\(\frac{du}{dx}=2x\)</span></td><td><span class="math-tex">\(\frac{dv}{dx}=3\)</span></td></tr></tbody></table><p> <span class="math-tex">\(\frac { dy }{ dx } =\frac { (3x-2)\cdot (2x) -(x^2+1)\cdot 3}{ (3x-2)^{ 2 } } \)</span></p><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { 6x^2-4x -3x^2-3}{ (3x-2)^{ 2 } } \)</span></p><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { 3x^2-4x -3}{ (3x-2)^{ 2 } } \)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> <hr class="hidden-separator"> <p>Here's a quiz that mainly practises the <em><strong>Product and Quotient Rule</strong></em></p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#7ba621fa"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="7ba621fa"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Mixed SL Differentiation 3 <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-385-2405" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Differentiate <span class="math-tex">\(\frac{2x}{\sqrt{1-x^2}}\)</span></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac{2}{ (1-x^2)^{\frac{3}{2}} }\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{2-4x^2}{ (1-x^2)^{\frac{3}{2}} }\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{2-4x^2}{ (1-x^2)^{2}}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { 2\sqrt { 1-x^{ 2 } } +\frac { 4x^{ 2 } }{ \sqrt { 1-x^{ 2 } } } }{ 1-x^{ 2 } } \)</span></label></p></div><div class="q-explanation"><p>We need to use the Quotient Rule</p><p><span class="math-tex">\(y=\frac { u }{ v } \\ \frac { dy }{ dx } =\frac { v\cdot \frac { du }{ dx } -u\cdot \frac { dv }{ dx } }{ v^{ 2 } } \)</span></p><hr class="hidden-separator"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>u = 2x</td><td><span class="math-tex">\(v=\sqrt{1-x^2}\\v=(1-x^2)^{0.5}\)</span></td></tr><tr><td><span class="math-tex">\(\frac { du }{ dx } =2\)</span></td><td><p><span class="math-tex">\(\frac { dv }{ dx } =0.5(-2x)(1-x^2)^{-0.5}\)</span></p><p><span class="math-tex">\(\frac { dv }{ dx } =\frac{-x}{\sqrt{1-x^2}}\)</span></p></td></tr></tbody></table><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { \sqrt{1-x^2}\cdot 2-2x\cdot \frac{-x}{\sqrt{1-x^2}} }{ 1-x^2 }\)</span></p><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { \frac{2(1-x^2)}{\sqrt{1-x^2}}+ \frac{2x^2}{\sqrt{1-x^2}} }{ 1-x^2 }\)</span></p><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { \frac{2-2x^2+2x^2}{\sqrt{1-x^2}}}{ 1-x^2 }\)</span></p><p><span class="math-tex">\(\frac { dy }{ dx } =\frac{2}{ (1-x^2)^{\frac{3}{2}} }\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Differentiate <span class="math-tex">\(e^{2x}lnx\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(2e^xlnx\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(e^{ 2x }(\frac { 1 }{ x } +2lnx)\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(2e^{ 2x }lnx+\frac { { e^{ 2x } } }{ lnx } \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(e^{ 2x }lnx+\frac { { e^{ 2x } } }{ x } \)</span></label></p></div><div class="q-explanation"><p>We need to use the product rule</p><p><span class="math-tex">\(y=uv\\ \frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}\)</span></p><hr class="hidden-separator"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><span class="math-tex">\(u=e^{2x}\)</span></td><td>v=lnx</td></tr><tr><td><span class="math-tex">\(\frac { du }{ dx } =2e^{2x}\)</span></td><td><span class="math-tex">\(\frac { dv }{ dx } =\frac{1}{x}\)</span></td></tr></tbody></table><p><span class="math-tex">\(\frac { dy }{ dx } =e^{ 2x }\frac { 1 }{ x } +lnx\cdot 2e^{ 2x }\)</span></p><p><span class="math-tex">\(\frac { dy }{ dx } =e^{ 2x }(\frac { 1 }{ x } +2lnx)\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Differentiate <span class="math-tex">\(e^xlnx^2\)</span></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac { 2e^x }{ x } + 2e^{ x }lnx\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(2e^{ x }lnx\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(e^ x(\frac { 1 }{ x } +2lnx)\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { e^{ x } }{ x^ 2 } +e^{ x }lnx^2\)</span></label></p></div><div class="q-explanation"><p>We need to use the product rule</p><p><span class="math-tex">\(y=uv\\ \frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}\)</span></p><hr class="hidden-separator"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><span class="math-tex">\(u=e^{x}\)</span></td><td><p>v=lnx²</p><p>v=2lnx</p></td></tr><tr><td><span class="math-tex">\(\frac { du }{ dx } =e^{x}\)</span></td><td><span class="math-tex">\(\frac { dv }{ dx } =2\cdot\frac{1}{x}\)</span></td></tr></tbody></table><p><span class="math-tex">\(\frac { dy }{ dx } =e^{ x }\frac { 2 }{ x } + 2e^{ x }lnx\)</span></p><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { 2e^x }{ x } + 2e^{ x }lnx\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Differentiate <span class="math-tex">\(\frac { e^{ { 2x } } }{ lnx } \)</span></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac { e^{ 2x }(2lnx -\frac { 1 }{ x }) }{ (lnx)^{ 2 } } \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { e^{ 2x }(xlnx-1) }{ x(lnx)^{ 2 } } \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { e^{ 2x }-2x{ e }^{ 2x }lnx }{ x(lnx)^{ 2 } } \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { e^{ 2x }(2lnx-1) }{ x(lnx)^{ 2 } } \)</span></label></p></div><div class="q-explanation"><p>We need to use the Quotient Rule</p><p><span class="math-tex">\(y=\frac { u }{ v } \\ \frac { dy }{ dx } =\frac { v\cdot \frac { du }{ dx } -u\cdot \frac { dv }{ dx } }{ v^{ 2 } } \)</span></p><hr class="hidden-separator"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>u = <span class="math-tex">\(e^{2x}\)</span></td><td>v = lnx</td></tr><tr><td><span class="math-tex">\(\frac { du }{ dx } =2e^{2x}\)</span></td><td><p><span class="math-tex">\(\frac { dv }{ dx } =\frac{1}{x}\)</span></p></td></tr></tbody></table><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { lnx\cdot 2e^{ 2x }-e^{ 2x }\cdot \frac { 1 }{ x } }{ (lnx)^{ 2 } } \)</span></p><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { e^{ 2x }(2lnx -\frac { 1 }{ x }) }{ (lnx)^{ 2 } } \)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Differentiate <span class="math-tex">\(\frac { { e }^{ 3x } }{ ln2x } \)</span></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac { e^{ 3x }(3ln2x-\frac { 1 }{ x } ) }{ (ln2x)^{ 2 } } \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { 3x{ e }^{ 3x }ln2x-e^{ 3x } }{ (ln2x)^{ 2 } } \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { 3x{ e }^{ 3x }ln2x-e^{ 3x } }{ x(ln2x)^{ 2 } } \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { e^{ 3x }(3ln2x-1) }{ 2xln2x } \)</span></label></p></div><div class="q-explanation"><p>We need to use the Quotient Rule</p><p><span class="math-tex">\(y=\frac { u }{ v } \\ \frac { dy }{ dx } =\frac { v\cdot \frac { du }{ dx } -u\cdot \frac { dv }{ dx } }{ v^{ 2 } } \)</span></p><hr class="hidden-separator"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>u = <span class="math-tex">\(e^{3x}\)</span></td><td>v = ln2x</td></tr><tr><td><span class="math-tex">\(\frac { du }{ dx } =3e^{3x}\)</span></td><td><p><span class="math-tex">\(\frac { dv }{ dx } =\frac{1}{x}\)</span></p></td></tr></tbody></table><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { ln2x\cdot 3e^{ 3x }-e^{ 3x }\cdot \frac { 1 }{ x } }{ (ln2x)^{ 2 } } \)</span></p><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { e^{ 3x }(3ln2x-\frac { 1 }{ x } ) }{ (ln2x)^{ 2 } } \)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Differentiate <span class="math-tex">\(\frac { ln3x }{ { e }^{ 2x } } \)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { { e }^{ 2x }-2x{ e }^{ 2x }ln3x }{ { e }^{ 2x } } \)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac { 1-2xln3x }{ { xe }^{ 2x } }\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { \frac { { e }^{ 2x } }{ x } -2{ e }^{ 2x }ln3x }{ { e }^{ 4x^{ 2 } } } \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { { e }^{ 2x }-2x{ e }^{ 2x }ln3x }{ { xe }^{ 4x^{ 2 } } } \)</span></label></p></div><div class="q-explanation"><p>We need to use the Quotient Rule</p><p><span class="math-tex">\(y=\frac { u }{ v } \\ \frac { dy }{ dx } =\frac { v\cdot \frac { du }{ dx } -u\cdot \frac { dv }{ dx } }{ v^{ 2 } } \)</span></p><hr class="hidden-separator"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>u = ln3x</td><td>v = <span class="math-tex">\(e^{2x}\)</span></td></tr><tr><td><span class="math-tex">\(\frac { du }{ dx } =\frac{1}{x}\)</span></td><td><p><span class="math-tex">\(\frac { dv }{ dx } =2e^{2x}\)</span></p></td></tr></tbody></table><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { e^{ 2x }\cdot \frac { 1 }{ x } -2e^{ 2x }\cdot ln3x }{ ({ e }^{ 2x })^{ 2 } } \)</span></p><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { e^{ 2x }(\frac { 1 }{ x } -2ln3x) }{ ({ e }^{ 2x })^{ 2 } } \)</span></p><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { \frac { 1 }{ x } -2ln3x }{ { e }^{ 2x } } \)</span></p><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { \frac { 1 }{ x } -\frac { 2xln3x }{ x } }{ { e }^{ 2x } } \)</span></p><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { 1-2xln3x }{ { xe }^{ 2x } } \)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Differentiate <span class="math-tex">\(\frac { { e }^{ x } }{ tanx } \)</span></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac { { e^xtanx }-e^{ x }sec²x }{ (tanx)^{ 2 } } \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { { e^x(tanx+sec²x)} }{ (tanx)^{ 2 } } \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { { e^xtanx-sec²x} }{ (tanx)^{ 2 } } \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { { e^x(sec²x-tanx)} }{ (tanx)^{ 2 } } \)</span></label></p></div><div class="q-explanation"><p>We need to use the Quotient Rule</p><p><span class="math-tex">\(y=\frac { u }{ v } \\ \frac { dy }{ dx } =\frac { v\cdot \frac { du }{ dx } -u\cdot \frac { dv }{ dx } }{ v^{ 2 } } \)</span></p><hr class="hidden-separator"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>u = <span class="math-tex">\(e^{x}\)</span></td><td>v = tanx</td></tr><tr><td><span class="math-tex">\(\frac { du }{ dx } =e^{x}\)</span></td><td><p><span class="math-tex">\(\frac { dv }{ dx } =sec²x\)</span></p></td></tr></tbody></table><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { { tanx }\cdot e^{ x }-e^{ x }sec²x }{ (tanx)^{ 2 } } \)</span></p><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { { e^xtanx }-e^{ x }sec²x }{ (tanx)^{ 2 } } \)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(f(x) = 2\sqrt{x}\)</span> , then <span class="math-tex">\(f''(x)=ax^b\)</span></p><p>Work out <strong><em>a</em></strong> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="-0.5"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="-1.5"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(f(x)=2x^{0.5}\)</span></p><p><span class="math-tex">\(f'(x)=2(0.5)x^{-0.5}=x^{-0.5}\)</span></p><p><span class="math-tex">\(f''(x)=-0.5x^{-1.5}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> If <span class="math-tex">\(f(x)=\frac { { x }^{ 3 } }{ { x }^{ 2 }+1 } \)</span>, find the gradient of the function when x = -1</p></div><div class="q-answer"><p><span class="math-tex">\(f'(-1)\)</span> = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"><p>We need to use the Quotient Rule</p><p><span class="math-tex">\(f(x)=\frac { g(x) }{ h(x) } \\ f'(x)=\frac { h(x)\cdot g'(x)-g(x)\cdot h'(x) }{ { \left[ h(x) \right] }^{ 2 } } \)</span></p><hr class="hidden-separator"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>g(x) = <span class="math-tex">\(x^3\)</span></td><td>h(x) = x²+1</td></tr><tr><td>g'(x) = 3x²</td><td><p>h'(x) = 2x</p></td></tr></tbody></table><p><span class="math-tex">\(f'(x)=\frac { (x²+1)(3x²)-{ x }^{ 3 }(2x) }{ { (x²+1) }^{ 2 } } \)</span></p><p><span class="math-tex">\(f'(-1)=\frac { ((-1)²+1)(3(-1)²)-{ (-1) }^{ 3 }(2(-1)) }{ {((-1)²+1 ) }^{ 2 } } \)</span></p><p><span class="math-tex">\(f'(-1)=\frac { (2)(3)-{ (-1) }(-2) }{ { (2) }^{ 2 } } \)</span></p><p><span class="math-tex">\(f'(-1)=\frac { 4 }{ 4 } \)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> The graph of the function <span class="math-tex">\(f(x)=\frac{lnx}{x}\)</span>, x>0 has a maximum value at x = <strong><em>a</em></strong></p><p>Work out the value of <strong><em>a</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="e"> <span class="review"></span></p></div><div class="q-explanation"><p>The local maximum of a function occurs when the gradient = 0</p><p>We need to use the Quotient Rule to find the gradient function</p><p><span class="math-tex">\(f(x)=\frac { g(x) }{ h(x) } \\ f'(x)=\frac { h(x)\cdot g'(x)-g(x)\cdot h'(x) }{ { \left[ h(x) \right] }^{ 2 } } \)</span></p><hr class="hidden-separator"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>g(x) = lnx</td><td>h(x) = x</td></tr><tr><td><span class="math-tex">\(g'(x) =\frac{1}{x}\)</span></td><td><p>h'(x) = 1</p></td></tr></tbody></table><p><span class="math-tex">\(f'(x) =\frac { { x }\cdot \frac { 1 }{ x } -lnx }{ x² } \)</span></p><p><span class="math-tex">\(f'(x)=\frac { 1-lnx }{ x² } \)</span></p><hr class="hidden-separator"><p>Solve <span class="math-tex">\(f'(x) = 0\)</span></p><p><span class="math-tex">\(\frac { 1-lnx }{ x² } =0\)</span></p><p>1 - lnx = 0 , since</p><p>lnx = 1</p><p>x = e</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> <hr class="hidden-separator"> <p> Here's a quiz that mainly practises the <em><strong>Related Rates of Change</strong></em></p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#e02eddd6"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="e02eddd6"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Related Rates of Change <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-603-2405" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>The rate at which the area, <strong><em>A</em></strong>, of a square of length <strong><em>x</em></strong> is increasing is given by <span class="math-tex">\(\large \frac{dA}{dt}\)</span></p><p>Which of the following will find the rate of increase in the length of the side, <span class="math-tex">\(\large \frac{dx}{dt}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{\frac{dA}{dt}}{x^2}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large 2x\frac{dA}{dt}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large x^2\frac{dA}{dt}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large \frac{\frac{dA}{dt}}{2x}\)</span></label></p></div><div class="q-explanation"><p>This is a related rates of change question. We use the chain rule</p><p><span class="math-tex">\(\large \frac{dA}{dt}=\frac{dA}{dx} \cdot \frac{dx}{dt}\)</span></p><p>The area of the square is given by the formula</p><p><span class="math-tex">\(\large A=x^2\)</span></p><p>We differentiate with respect to x to find <span class="math-tex">\(\large \frac{dA}{dx}\)</span></p><p><span class="math-tex">\(\large \frac{dA}{dx}=2x\)</span></p><hr class="hidden-separator"><p>We substitute this into the chain rule and rearrange</p><p><span class="math-tex">\(\large \frac{dA}{dt}=2x \cdot \frac{dx}{dt}\)</span></p><p><span class="math-tex">\(\large \frac{\frac{dA}{dt}}{2x}= \frac{dx}{dt}\)</span></p><p> </p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The rate at which the radius, <strong><em>r</em></strong>, of a sphere is increasing is given by <span class="math-tex">\(\large \frac{dr}{dt}\)</span></p><p>Which of the following will find the rate of increase of the volume, <span class="math-tex">\(\large \frac{dV}{dt}\)</span></p><p style="text-align: right;">The volume of a sphere is given by <span class="math-tex">\(\large V= \frac{4}{3}\pi r^3 \)</span></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large 4\pi r^2 \frac{dr}{dt}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{4}{3}\pi r^3 \frac{dr}{dt}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{\frac{dr}{dt}}{\frac{4}{3}\pi r^3}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{4\pi r^2}{\frac{dr}{dt}}\)</span></label></p></div><div class="q-explanation"><p>This is a related rates of change question. We use the chain rule</p><p><span class="math-tex">\(\large \frac{dV}{dt}=\frac{dV}{dr} \cdot \frac{dr}{dt}\)</span></p><p>The volume of a sphere is given by</p><p><span class="math-tex">\(\large V= \frac{4}{3}\pi r^3 \)</span></p><p>We differentiate with respect to r to find <span class="math-tex">\(\large \frac{dV}{dr}\)</span></p><p><span class="math-tex">\(\large \frac{dV}{dr}=4\pi r^2\)</span></p><hr class="hidden-separator"><p>We substitute this into the chain rule</p><p><span class="math-tex">\(\large \frac{dV}{dt}=4\pi r^2 \cdot \frac{dr}{dt}\)</span></p><p> </p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The radius, <strong><em>r</em></strong>, of a circle is increasing at the rate of <span class="math-tex">\(\large \frac{3}{\pi} \ cms^{-1}\)</span></p><p>Find the rate at which the circumference, <strong><em>C</em></strong>, is increasing</p></div><div class="q-answer"><p><span class="math-tex">\(\large \frac{dC}{dt}\)</span>= <input type="text" style="height: auto;" data-c="6"> <span class="review"></span> <span class="math-tex">\(\large cms^{-1}\)</span></p></div><div class="q-explanation"><p>This is a related rates of change question. We use the chain rule</p><p><span class="math-tex">\(\large \frac{dC}{dt}=\frac{dC}{dr} \cdot \frac{dr}{dt}\)</span></p><p>The circumference of a circle is given by the formula</p><p><span class="math-tex">\(\large C=2 \pi r\)</span></p><p>We differentiate with respect to r to find <span class="math-tex">\(\large \frac{dC}{dr}\)</span></p><p><span class="math-tex">\(\large \frac{dC}{dr}=2\pi\)</span></p><hr class="hidden-separator"><p>We substitute this into the chain rule</p><p><span class="math-tex">\(\large \frac{dC}{dt}=2\pi \cdot \frac{3}{\pi} =6\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The radius, <strong><em>r</em></strong>, of a circle is increasing at the rate of <span class="math-tex">\(\large \frac{1}{\pi} \ cms^{-1}\)</span></p><p>Find the rate at which the circumference, <strong><em>A</em></strong>, is increasing when <strong><em>r</em></strong> = 5cm</p></div><div class="q-answer"><p><span class="math-tex">\(\large \frac{dA}{dt}\)</span>= <input type="text" style="height: auto;" data-c="10"> <span class="review"></span> <span class="math-tex">\(\large cm^2s^{-1}\)</span></p></div><div class="q-explanation"><p>This is a related rates of change question. We use the chain rule</p><p><span class="math-tex">\(\large \frac{dA}{dt}=\frac{dA}{dr} \cdot \frac{dr}{dt}\)</span></p><p>The area of a circle is given by the formula</p><p><span class="math-tex">\(\large A=\pi r^2\)</span></p><p>We differentiate with respect to r to find <span class="math-tex">\(\large \frac{dA}{dr}\)</span></p><p><span class="math-tex">\(\large \frac{dA}{dr}=2\pi r\)</span></p><p>When <strong><em>r</em></strong> = 5cm, <span class="math-tex">\(\large \frac{dA}{dr}=10\pi \)</span></p><hr class="hidden-separator"><p>We substitute this into the chain rule</p><p><span class="math-tex">\(\large \frac{dA}{dt}=10\pi \cdot \frac{1}{\pi} =10\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The volume of a cube is increasing at a rate of 30<span class="math-tex">\(cm^3s^{-1}\)</span></p><p>Find the rate at which the sides of the cube, <strong><em>x</em></strong>, are increasing when the lengths of the sides are 10cm</p></div><div class="q-answer"><p><span class="math-tex">\(\large \frac{dx}{dt}\)</span>= <input type="text" style="height: auto;" data-c="0.1"> <span class="review"></span> <span class="math-tex">\(\large cms^{-1}\)</span></p></div><div class="q-explanation"><p>This is a related rates of change question. We use the chain rule</p><p><span class="math-tex">\(\large \frac{dV}{dt}=\frac{dV}{dx} \cdot \frac{dx}{dt}\)</span></p><p>The volume of the cube is given by the formula</p><p><span class="math-tex">\(\large V=x^3\)</span></p><p>We differentiate with respect to x to find <span class="math-tex">\(\large \frac{dV}{dx}\)</span></p><p><span class="math-tex">\(\large \frac{dV}{dx}=3x^2\)</span></p><p>When x = 10, <span class="math-tex">\(\large \frac{dV}{dx}=300\)</span></p><hr class="hidden-separator"><p>We substitute this into the chain rule</p><p><span class="math-tex">\(\large30=300 \cdot \frac{dx}{dt}\)</span></p><p><span class="math-tex">\(\large \frac{dx}{dt}\)</span>= 0.1</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>A spherical balloon is deflating at a rate of <span class="math-tex">\(\large 160\pi \ cm^3s^{-1}\)</span></p><p>At what rate is the radius of the balloon decreasing when the radius, r = 4cm ?</p><p style="text-align: right;">The volume of a sphere is given by <span class="math-tex">\(\large V= \frac{4}{3}\pi r^3 \)</span></p></div><div class="q-answer"><p><span class="math-tex">\(\large \frac{dr}{dt}\)</span>= <input type="text" style="height: auto;" data-c="2.5"> <span class="review"></span> <span class="math-tex">\(\large cms^{-1}\)</span></p></div><div class="q-explanation"><p>This is a related rates of change question. We use the chain rule</p><p><span class="math-tex">\(\large \frac{dV}{dt}=\frac{dV}{dr} \cdot \frac{dr}{dt}\)</span></p><p>The volume of a sphere is given by <span class="math-tex">\(\large V= \frac{4}{3}\pi r^3 \)</span></p><p>We differentiate with respect to r to find <span class="math-tex">\(\large \frac{dV}{dr}\)</span></p><p><span class="math-tex">\(\large \frac{dV}{dr}=4\pi r^2\)</span></p><p>when the radius, r = 4cm , <span class="math-tex">\(\large \frac{dV}{dr}=4\pi \times 4^2=64 \pi\)</span></p><hr class="hidden-separator"><p>We substitute this into the chain rule</p><p><span class="math-tex">\(\large160\pi=64\pi\cdot \frac{dr}{dt}\)</span></p><p><span class="math-tex">\(\large \frac{dr}{dt}=\frac{160\pi}{64\pi}=2.5\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> <hr class="hidden-separator"> <p>Here's a quiz that practises differentiating using a <strong>graphical display calculator</strong></p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#757294f2"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="757294f2"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Using GDC for Differentiation <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-601-2405" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> Use your graphic display calculator to solve this question</p><p>Let <span class="math-tex">\(\large f(x)=3^{-x}+4x-5\)</span></p><p>The graph of <strong><em>f</em></strong> has a horizontal tangent at the point where x = <strong><em>a</em></strong></p><p>Find the value of <strong><em>a</em></strong></p><p><strong><em>Give your answer to 3 s.f. </em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="-1.18"> <span class="review"></span></p></div><div class="q-explanation"><p>A horizontal tangent has a gradient = 0</p><p>You need to find the value of x for which <span class="math-tex">\(\large f'(x)=0\)</span></p><p><img alt="" src="../../files/differentiation/using-gdc/q1ans.png" style="width: 268px; height: 154px;"></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> Use your graphic display calculator to solve this question</p><p>Let <span class="math-tex">\(\large f(x)=\ln 2x +2^x\)</span></p><p>The tangent to the graph of <strong><em>f</em></strong> at <strong><em>x=1</em></strong> intersects the y axis at y = <strong><em>a</em></strong></p><p>Find the value of <strong><em>a</em></strong></p><p><strong><em>Give your answer to 3 s.f. </em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="0.307"> <span class="review"></span></p></div><div class="q-explanation"><p>Use your graphical display calculator to sketch the graph of the tangent to the function <strong><em>f</em></strong> at x = 1</p><p> <img alt="" src="../../files/differentiation/using-gdc/q2ans.png" style="width: 268px; height: 154px;"></p><p>The equation of the tangent is found to be y = 2.39x + 0.307</p><p>Hence this intersects the y axis when x = 0</p><p>To 3 significant figures, this is y = 0.307</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> Use your graphic display calculator to solve this question</p><p>Let <span class="math-tex">\(\large f(x)=\frac{2x}{2x-5}\ , \ x\neq\frac{5}{2}\)</span></p><p>The normal to the graph of <em><strong>f </strong></em>at x = 0<em><strong> </strong></em>intersects the graph of <strong><em>f</em></strong> again at x = <strong><em>a</em></strong></p><p>Find <strong><em>a</em></strong></p><p><strong><em>Give the exact answer</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="2.9"> <span class="review"></span></p></div><div class="q-explanation"><p>Use your graphical display calculator to sketch the graph of the normal to the function <strong><em>f</em></strong> at x = 0</p><p><img alt="" src="../../files/differentiation/using-gdc/q3ans1.png" style="width: 268px; height: 154px;"></p><p>Plot <strong><em>f</em></strong> and the equation of the normal on the same axes and find the point of intersection</p><p><img alt="" src="../../files/differentiation/using-gdc/q3ans2.png" style="width: 268px; height: 154px;"></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> Use your graphic display calculator to solve this question</p><p>Let <span class="math-tex">\(\large f'(x)=\sin2 x\cos3 x\ ,\ 0\le x\le1\)</span></p><p>The graph of <strong><em>f</em></strong> has two points of inflexion at x = a and x = b, where a < b<b< p=""> </b<></p><p>Find the value of <strong><em>a</em></strong></p><p><strong><em>Give your answer to 3 s.f. </em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="0.275"> <span class="review"></span></p></div><div class="q-explanation"><p>Make sure that your calculator is in radians mode</p><p>Notice that the domain for <span class="math-tex">\(\large f'\)</span> exists for <span class="math-tex">\(0\le x\le1\)</span></p><p>The points of inflexion for the graph of <strong><em>f</em></strong><em> </em>exist where <span class="math-tex">\(\large f''(x)=0\)</span></p><p>You can plot the graph of <span class="math-tex">\(\large f'\)</span> and find the x coordinate of the first stationary point for <span class="math-tex">\(\large f'\)</span>, in this case, a local maximum</p><p><img alt="" src="../../files/differentiation/using-gdc/q4ans.png" style="width: 268px; height: 154px;"></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> Use your graphic display calculator to solve this question</p><p>Let <span class="math-tex">\(\large f(x)=x^4-2x^2+x\)</span></p><p>The following diagram shows the graph of <strong><em>f</em></strong></p><p><img alt="" src="../../files/differentiation/using-gdc/q5.png" style="width: 400px; height: 396px;"></p> <p>Find the x coordinates of the point A</p></div><div class="q-answer"><p><strong><em>x</em></strong> = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"><p>Use your calculator to find the root of the equation f(x) = 0</p><p>Ensure that you find the correct root</p><p><img alt="" src="../../files/differentiation/using-gdc/q5ans.png" style="width: 268px; height: 154px;"></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> Use your graphic display calculator to solve this question</p><p>Let <span class="math-tex">\(\large f(x)=\ln (x)-3x\ , \ x>0\)</span></p><p><em>Solve</em><strong><em> <span class="math-tex">\(\large f'(x)=f''(x)\)</span></em></strong></p><p> </p><p><strong><em>Give your answer to 3 s.f. </em></strong></p></div><div class="q-answer"><p><strong><em>x</em></strong> = <input type="text" style="height: auto;" data-c="-0.434"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large f(x)=\ln (x)-3x\ , \ x>0\)</span></p><p><span class="math-tex">\(\large f'(x)=\frac{1}{x}-3\)</span></p><p><span class="math-tex">\(\large f''(x)=-\frac{1}{x^2}\)</span></p><p> </p><p>Solve <span class="math-tex">\(\large \frac{1}{x}-3=-\frac{1}{x^2}\)</span></p><p>You can do this by plotting the graphs of <span class="math-tex">\(\large y=\frac{1}{x}-3\)</span> and <span class="math-tex">\(\large y=-\frac{1}{x^2}\)</span> and finding the x coordinate of the point of intersection</p><p><img alt="" src="../../files/differentiation/using-gdc/q6ans.png" style="width: 268px; height: 154px;"></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> Use your graphic display calculator to solve this question</p><p>Let <span class="math-tex">\(\large f(x)=x^3e^{-2x}\)</span></p>Find the x coordinate of the point of intersection of the tangents to the graph of <strong><em>f </em></strong>at x = 1 and x = 2<p> </p><p><strong><em>Give your answer to 2 s.f. </em></strong></p></div><div class="q-answer"><p><strong><em>x</em></strong> = <input type="text" style="height: auto;" data-c="1.4"> <span class="review"></span></p></div><div class="q-explanation"><p>Note that the answer only needs to be given to 2 s.f.</p><p>Use your graphical display calculator to sketch the graph of the tangent to the function <strong><em>f</em></strong> at x = 1 and also for x = 2</p><p><img alt="" src="../../files/differentiation/using-gdc/q7ans.png" style="width: 686px; height: 177px;"></p><p>Plot these two tangents and find the point of intersection</p><p><img alt="" src="../../files/differentiation/using-gdc/q7ans2.png" style="width: 268px; height: 154px;"></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> Use your graphic display calculator to solve this question</p><p>Let <span class="math-tex">\(\large f(x)=x^4-2e^x\)</span></p><p>The following diagram shows the graph of <strong><em>f</em></strong></p><p><strong><em><img alt="" src="../../files/differentiation/using-gdc/q8.png" style="width: 400px; height: 362px;"></em></strong></p><p><strong><em>A </em></strong>is a point of inflexion</p><p>The coordinates of <em><strong>A</strong></em> are (<em><strong>a</strong></em>,<em><strong>f(a)</strong></em>)</p><p>Find the value of <strong><em>a</em></strong></p><p><strong><em>Give your answer to 3 s.f. </em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="0.533"> <span class="review"></span></p></div><div class="q-explanation"><p>To find a point of inflexion, we need to solve <span class="math-tex">\(\large f''(x)=0\)</span></p><p>This is also a stationary point of the function <span class="math-tex">\(\large f'(x)\)</span></p><p>We can differentiate the function to find <span class="math-tex">\(\large f'(x)=4x^3-2e^x\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> Use your graphic display calculator to solve this question</p><p>Let <span class="math-tex">\(\large f(x)=sin(e^x) \ , \ 0\le x \le 2\)</span></p><p>The curve of <em><strong>f</strong></em> is concave up on the interval<strong><em> a < x < b</em></strong></p><p>Find the values of <strong><em>a </em></strong>and <strong><em>b</em></strong></p><p><strong><em>Give your answer to 3 s.f. </em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="1.23"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="1.86"> <span class="review"></span></p></div><div class="q-explanation"><p>The curve of a function is concave up when <span class="math-tex">\(\large f''(x)>0\)</span></p><p>This will occur between two points of inflexion</p><p><img alt="" src="../../files/differentiation/using-gdc/q9ans.png" style="width: 400px; height: 361px;"></p><p>Differentiate <span class="math-tex">\(\large f(x)=sin(e^x) \)</span> using the Chain Rule</p><p><span class="math-tex">\(\large f'(x)=e^xcos(e^x) \)</span></p><p>We can find the points of inflexion for <span class="math-tex">\(f(x)\)</span> by finding the stationary points for <span class="math-tex">\(f'(x)\)</span></p><p><img alt="" src="../../files/differentiation/using-gdc/q9ans2.png" style="width: 681px; height: 177px;"></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> Use your graphic display calculator to solve this question</p><p>Let <span class="math-tex">\(\large f(x)=2x \ln x \ ,\ x>0\)</span></p><p>Points <em><strong>P</strong></em>(1,0) and <em><strong>Q</strong></em> are on the curve of <strong><em>f</em></strong>.</p><p>The tangent to the curve of <strong><em>f</em></strong> at <em><strong>P</strong></em> is perpendicular to the tangent to the curve at <em><strong>Q</strong></em>.</p><p>Find the x coordinates of <em><strong>Q</strong></em>.</p> <p><strong><em>Give your answer to 3 s.f. </em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="0.287"> <span class="review"></span></p></div><div class="q-explanation"><p>You do not need to plot a graph of the curve and the tangents, but it may help you visualise the problem.</p><p>Plot the graph of the function and the tangent to the curve at the point x = 1</p><p>To get a true picture of the graph, you may want to plot the graph so that the axes are 1:1 (so that a right angle actually looks like a right angle).</p><p><img alt="" src="../../files/differentiation/using-gdc/q10ans1.png" style="width: 268px; height: 154px;"></p><p>You will notice that the gradient of this tangent = 2</p><p>You need to find the x coordinates of the graph for which the curve has a gradient = <span class="math-tex">\(-\frac{1}{2}\)</span></p><p>Differentiate the function using the product rule</p><p><span class="math-tex">\(\large f'(x)=2 \ln x+2 \)</span></p><p>Solve <span class="math-tex">\(\large 2 \ln x+2 =-\frac{1}{2}\)</span></p><p>You can use the equation solver or a graphical approach.</p><p>Here is what the two tangents look like</p><p><img alt="" src="../../files/differentiation/using-gdc/q10ans2.png" style="width: 268px; height: 154px;"></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> <hr class="hidden-separator"> <p>Here's a quiz that practises <em><strong>Graphs and the Gradient Function</strong></em></p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#d1186194"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="d1186194"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Mixed SL Differentiation 4 <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-386-2405" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>The diagram shows the graph of y = f(x)</p><p><img alt="" src="../../files/differentiation/mixed-differentiation/quiz4-graphs/q1.png" style="width: 400px; height: 400px;"></p><p>Which of the following is true?</p><p>Select <strong>ALL</strong> correct answers</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(f'(b)=0\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(f''(b)<0\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(f'(c)>0\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> f(x) is increasing from a to b</label></p></div><div class="q-explanation"><p>For a local maxima</p><ul><li><span class="math-tex">\(f'(x)=0\)</span></li><li><span class="math-tex">\(f''(x)<0\)</span></li></ul></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The diagram shows the graph of y = f(x)</p><p><img alt="" src="../../files/differentiation/mixed-differentiation/quiz4-graphs/q2.png" style="width: 300px; height: 300px;"></p><p>Which of the following is the correct graph of <span class="math-tex">\(y = f'(x)\)</span></p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>A</td><td><img alt="" src="../../files/differentiation/mixed-differentiation/quiz4-graphs/q2a.png" style="width: 300px; height: 300px;"></td></tr><tr><td>B</td><td><img alt="" src="../../files/differentiation/mixed-differentiation/quiz4-graphs/q2b.png" style="width: 300px; height: 300px;"></td></tr><tr><td>C</td><td><img alt="" src="../../files/differentiation/mixed-differentiation/quiz4-graphs/q2c.png" style="width: 300px; height: 300px;"></td></tr><tr><td>D</td><td><img alt="" src="../../files/differentiation/mixed-differentiation/quiz4-graphs/q2d.png" style="width: 300px; height: 300px;"></td></tr></tbody></table></div><div class="q-answer"><p><label class="radio" style=" float: left; margin-right: 40px; "><input class="c" type="radio"> C</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> A</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> B</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> D</label></p></div><div class="q-explanation"><p><img alt="" src="../../files/differentiation/mixed-differentiation/quiz4-graphs/q2explanation.png" style="width: 600px; height: 538px;"></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The diagram shows the graph of y = f(x)</p><p><img alt="" src="../../files/differentiation/mixed-differentiation/quiz4-graphs/q3.png" style="width: 400px; height: 400px;"></p><p>Which of the following is true</p><p>Select <strong>ALL</strong> correct answers</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(f(a)=0\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(f'(b)=0\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(f''(d)=0\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(f'(c)>f'(e)\)</span></label></p></div><div class="q-explanation"><p>At local maxima <span class="math-tex">\(f'(x)\)</span>= 0 and <span class="math-tex">\(f''(x)<0\)</span></p><p>At local minima <span class="math-tex">\(f'(x)\)</span>= 0 and <span class="math-tex">\(f''(x)>0\)</span></p><p>Note that, gradient at c is negative and gradient at e is positive. Therefore, <span class="math-tex">\(f'(c)<f'(e)\)< span=""></f'(e)\)<></span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The diagram shows the graph of <span class="math-tex">\(y=f'(x)\)</span></p><p><img alt="" src="../../files/differentiation/mixed-differentiation/quiz4-graphs/q4.png" style="width: 400px; height: 270px;"></p><p>Which of the following is <strong>TRUE </strong>about <strong><span class="math-tex">\(y=f(x)\)</span></strong></p><p>Select <strong>ALL</strong> correct answers</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> f(x) has 2 local maxima</label></p><p><label class="checkbox"><input type="checkbox"> f(x) has two stationary points</label></p><p><label class="checkbox"><input class="c" type="checkbox"> f(x) has two points of inflexion</label></p><p><label class="checkbox"><input class="c" type="checkbox"> f(x) has a local minima at x = c</label></p></div><div class="q-explanation"><p>The diagram below shows the graph of the function y = f(x) in black</p><p><img alt="" src="../../files/differentiation/mixed-differentiation/quiz4-graphs/q4explanation.png" style="width: 400px; height: 368px;"></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The diagram shows the graph of <span class="math-tex">\(y=f'(x)\)</span></p><p><img alt="" src="../../files/differentiation/mixed-differentiation/quiz4-graphs/q5.png" style="width: 400px; height: 400px;"></p><p>Which of the following is <strong>TRUE</strong></p><p>Select <strong>ALL</strong> correct answers</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(f''(b)=0\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(f''(c)=0\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(f'(b)=0\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(f''(c)<0\)</span></label></p></div><div class="q-explanation"><p>The diagram below shows the graph of the function <span class="math-tex">\(y = f''(x)\)</span> in green</p><p><img alt="" src="../../files/differentiation/mixed-differentiation/quiz4-graphs/q5explanation2.png" style="width: 400px; height: 400px;"></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The diagram shows the graph of <span class="math-tex">\(y=f'(x)\)</span></p><p><img alt="" src="../../files/differentiation/mixed-differentiation/quiz4-graphs/q6.png" style="width: 400px; height: 400px;"></p><p>Which is the correct sketch of the graph of <span class="math-tex">\(y=f(x)\)</span></p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>A</td><td><img alt="" src="../../files/differentiation/mixed-differentiation/quiz4-graphs/q6a.png" style="width: 400px; height: 400px;"></td></tr><tr><td>B</td><td><img alt="" src="../../files/differentiation/mixed-differentiation/quiz4-graphs/q6b.png" style="width: 400px; height: 400px;"></td></tr><tr><td>C</td><td><img alt="" src="../../files/differentiation/mixed-differentiation/quiz4-graphs/q6c.png" style="width: 400px; height: 400px;"></td></tr><tr><td>D</td><td><img alt="" src="../../files/differentiation/mixed-differentiation/quiz4-graphs/q6d.png" style="width: 400px; height: 400px;"></td></tr></tbody></table></div><div class="q-answer"><p><label class="radio" style=" float: left; margin-right: 40px; "><input class="c" type="radio"> B</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> A</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> C</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> D</label></p></div><div class="q-explanation"><p>We are given the graph of the gradient function and asked to work out what the original function might look like</p><p><img alt="" src="../../files/differentiation/mixed-differentiation/quiz4-graphs/q6explanation.png" style="width: 400px; height: 423px;"></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The function f(x) has a local maxima at x = <em><strong>b</strong></em>.</p><p><em><strong>a</strong></em> is a value close to <em><strong>b</strong></em> such that <em><strong>a</strong></em><<em><strong>b</strong></em></p><p>Which of the following is true?</p><p>Select <strong>ALL </strong>correct answers</p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(f(b) = 0\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(f'(a)>0\) </span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(f''(b)>0\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(f'(b) = 0\)</span></label></p></div><div class="q-explanation"><p><img alt="" src="../../files/differentiation/mixed-differentiation/quiz4-graphs/q7.png" style="width: 400px; height: 400px;"></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The function f(x) has a local minima at x = <em><strong>b</strong></em>.</p><p><em><strong>a</strong></em> and <em><strong>c</strong></em> are values close to <em><strong>b</strong></em> such that <em><strong>a</strong></em><<em><strong>b</strong></em><<em><strong>c</strong></em></p><p>Which of the following is true?</p><p>Select <strong>ALL </strong>correct answers</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(f'(c)>0\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(f'(a)<0\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(f''(b)<0\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(f''(b)>0\)</span></label></p></div><div class="q-explanation"><p><img alt="" src="../../files/differentiation/mixed-differentiation/quiz4-graphs/q8.png" style="width: 400px; height: 400px;"></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The function f has a non-stationary point of inflexion at x = <em><strong>b</strong></em></p><p><strong><em>a</em></strong> and <strong><em>c</em></strong> are values close to <strong><em>b </em></strong> such that <strong><em>a</em></strong> < <strong><em>b</em></strong> < <strong><em>c</em></strong></p><p>Which of the following is true.</p><p>Select <strong>ALL</strong> correct answers.</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(f''(b)=0\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(f'(a)<0 \quad and \quad f'(c)>0\)</span></label> </p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(f'(b)=0\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(f(a)<f(b)<f(c) \quad or \quad f(a)>f(b)>f(c)\)</span></label><span class="math-tex"><span class="math-tex"><><f(c)><f(c)></f(c)></f(c)></span><><f(c)\)< span=""> <f(c)\)<></f(c)\)<></f(c)\)<></span></p></div><div class="q-explanation"><p>If there is a point of inflexion then <span class="math-tex">\(f''(x)=0\)</span></p><p>If it is a non-stationary point then <span class="math-tex">\(f'(x)\neq 0\)</span></p><p>The function must either be</p><ul><li>increasing from x=a to x=c, so <span class="math-tex"><span class="math-tex"><span class="math-tex">\(f(a)<f(b)<f(c)\)</span><><f(c)\)< span=""><><f(c)\)< span=""><f(c)\)<><f(c)\)<></f(c)\)<></f(c)\)<></f(c)\)<></f(c)\)<></span></span></li><li>decreasing from x=a to x=c, so <span class="math-tex">\(f(a)>f(b)>f(c)\)</span></li></ul><p>The sign of the gradient at x=a and x=c must be the same (they must be either both positive, or both negative).</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The following is true about a function <strong><em>f</em></strong></p><ul><li><span class="math-tex"><><c\)< span=""><><><><c< li=""><><><c<><c\)<></c\)<></c<></c<></c\)<></span>a < b < c</li><li>f(x) is increasing on the interval [a,c]</li><li><span class="math-tex">\(f'(x)<0\)</span> for <span class="math-tex">\(x < a\)</span></li><li><span class="math-tex">\(f''(b)=0\)</span></li><li><span class="math-tex">\(f'(c)=0\)</span><hr class="hidden-separator"></li></ul><p>Which of the following is TRUE?</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> There is a local minima at x = a</label></p><p><label class="checkbox"><input class="c" type="checkbox"> There is a point of inflexion at x = b</label></p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(f'(b)=0\)</span></label><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(f'(a)=0\)</span></label></p></div><div class="q-explanation"><p><img alt="" src="../../files/differentiation/mixed-differentiation/quiz4-graphs/q10explanation.png" style="width: 500px; height: 452px;"></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> <hr class="hidden-separator"> <p>Here's a quiz that practises <em><strong>Stationary Points and Equations of Tangents and Normals</strong></em></p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#9ff85a24"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="9ff85a24"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Mixed SL Differentiation 5 <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-390-2405" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Find the gradient of the curve <span class="math-tex">\(y = x^3-2x^2\)</span> where x= 1</p></div><div class="q-answer"><p>gradient = <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span></p></div><div class="q-explanation"><p>Differentiate <span class="math-tex">\(y = x^3-2x^2\)</span></p><p><span class="math-tex">\(\frac{dy}{dx}=3x^2-4x\)</span></p><p>when x = 1, <span class="math-tex">\(\frac{dy}{dx}=3(1)^2-4(1)=-1\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> The gradient of the curve y = sin3x at the point A is 1.5</p><p>The x cordinate of A is <span class="math-tex">\(\frac{\pi}{k}\)</span>such that <span class="math-tex">\(0<\frac{\pi}{k}<\frac{\pi}{2}\)</span></p><p>Work out the value of <strong><em>k</em></strong></p></div><div class="q-answer"><p><strong><em>k</em></strong> = <input type="text" style="height: auto;" data-c="9"> <span class="review"></span></p></div><div class="q-explanation"><p>y = sin3x,</p><p><span class="math-tex">\(\frac{dy}{dx}=3cos3x\)</span></p><p>We know that <span class="math-tex">\(\frac{dy}{dx}=\frac{3}{2}\)</span></p><p><span class="math-tex">\(3cos3x=\frac{3}{2}\)</span></p><p><span class="math-tex">\(cos3x=\frac{1}{2}\)</span></p><p>Arccos(0.5)=<span class="math-tex">\(\frac{\pi}{3}\)</span></p><p>3x = <span class="math-tex">\(\frac{\pi}{3}\)</span></p><p>x = <span class="math-tex">\(\frac{\pi}{9}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> What is the gradient of the normal to the curve <span class="math-tex">\(y = \frac{e^x}{x^2}\)</span> at x = 1</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(-\frac{1}{e}\)</span></label></p><p><label class="radio"><input type="radio"> e</label></p><p><label class="radio"><input type="radio"> -e</label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac{1}{e}\)</span></label></p></div><div class="q-explanation"><p>Differentiate <span class="math-tex">\(y = \frac{e^x}{x^2}\)</span></p><p>We need to use the Quotient Rule</p><p><span></span><span class="math-tex">\(y=\frac { u }{ v } \\ \frac { dy }{ dx } =\frac { v\cdot \frac { du }{ dx } -u\cdot \frac { dv }{ dx } }{ v^{ 2 } } \)</span><span></span></p><hr class="hidden-separator"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>u = <span class="math-tex">\(e^x\)</span></td><td>v = x²</td></tr><tr><td><span class="math-tex">\(\frac{du}{dx}=e^x\)</span></td><td><span class="math-tex">\(\frac{dv}{dx}=2x\)</span></td></tr></tbody></table><p><span></span> <span class="math-tex">\(\frac{dy}{dx}=\frac { x^{ 2 }e^{ x }-e^{ x }\cdot 2x }{ (x^{ 2 })^{ 2 } } \)</span></p><p>when x = 1 , <span class="math-tex">\(\frac{dy}{dx}=\frac { 1^{ 2 }e^{ 1 }-2e^{ 1 } }{ 1^{ 4 } } \)</span></p><p><span class="math-tex">\(\frac{dy}{dx}=\frac { e-2e }{ 1 } =-e\)</span></p><p>Gradient of tangent = -e</p><p>Gradient of normal = <span class="math-tex">\(\frac{-1}{-e}=\frac{1}{e}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Find the gradient of the tangent to <span class="math-tex">\(f(x)=\frac{1}{e^x}\)</span> at x=ln2</p></div><div class="q-answer"><p>gradient = <input type="text" style="height: auto;" data-c="-0.5"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(f(x)=e^{-x}\)</span></p><p><span class="math-tex">\(f'(x)=-e^{-x}\)</span></p><p><span class="math-tex">\(f'(x)= -\frac{1}{e^x}\)</span></p><p><span class="math-tex">\(f'(ln2)= -\frac{1}{e^{ln2}}=-\frac{1}{2}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Find the equation of the tangent to the curve y = xtanx at <span class="math-tex">\((\pi,0)\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(y=-\pi x-\pi^2\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(y=-\pi x+\pi^2\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(y=\pi x-\pi^2\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(y=\pi x+\pi^2\)</span></label></p></div><div class="q-explanation"><p>To differentiate y = xtanx, we need to use the Product Rule</p><p><span></span><span class="math-tex">\(y=uv\\ \frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}\)</span><span></span></p><hr class="hidden-separator"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>u = x</td><td>v = tanx</td></tr><tr><td><span class="math-tex">\(\frac{du}{dx}=1\)</span></td><td><span class="math-tex">\(\frac{dv}{dx}=sec²x\)</span></td></tr></tbody></table><p><span></span> <span class="math-tex">\(\frac{dy}{dx}=xsec²x+tanx\cdot1\)</span></p><p>When x = <span class="math-tex">\(\pi\)</span>, <span class="math-tex">\(\frac{dy}{dx}=\pi sec²(\pi)+tan(\pi)\)</span></p><p><span class="math-tex">\(\frac{dy}{dx}=\frac{\pi}{[ cosx(\pi)]^2}+tan(\pi)\)</span></p><p><span class="math-tex">\(\frac{dy}{dx}=\frac{\pi}{( -1)^2}+0=\pi\)</span></p><hr class="hidden-separator">Find the equation of the straight line with gradient = <span class="math-tex">\(\pi\)</span> that passes through the point <span class="math-tex">\((\pi,0)\)</span><p><span class="math-tex">\(\frac{y-0}{x-\pi}=\pi\)</span></p><p><span class="math-tex">\(y=\pi(x-\pi)\)</span></p><p><span class="math-tex">\(y=\pi x-\pi^2\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Find the equation of the normal to the curve <span class="math-tex">\(y=x^3lnx\)</span> at (1,0)</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> y = -x + 1</label></p><p><label class="radio"><input type="radio"> y = 4x - 4</label></p><p><label class="radio"><input type="radio"> 4y = -x + 1</label></p><p><label class="radio"><input type="radio"> y = x - 1</label></p></div><div class="q-explanation"><p>We need to use the product rule to differentiate <span class="math-tex">\(y=x^3lnx\)</span></p><p><span></span><span class="math-tex">\(y=uv\\ \frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}\)</span><span></span></p><hr class="hidden-separator"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>u = <span class="math-tex">\(x^3\)</span></td><td>v = lnx</td></tr><tr><td><span class="math-tex">\(\frac{du}{dx}=3x^2\)</span></td><td><span class="math-tex">\(\frac{dv}{dx}=\frac{1}{x}\)</span></td></tr></tbody></table><p><span></span> <span class="math-tex">\(\frac{dy}{dx}=x^3\cdot \frac{1}{x}+lnx\cdot3x^2\)</span></p><p><span class="math-tex">\(\frac{dy}{dx}=x^2+3x^2lnx\)</span></p><p><span class="math-tex">\(\frac{dy}{dx}=x^2(1+lnx)\)</span></p><p>When x = 1, <span class="math-tex">\(\frac{dy}{dx}=1(1+ln1)=1(1+0)=1\)</span></p><p>Gradient of tangent = 1</p><p>Gradient of normal = -1</p><hr class="hidden-separator"><p>Find the equation of the straight line with gradient = -1 which passes through the point (1,0)</p><p><span class="math-tex">\(\frac{y-0}{x-1}=-1\)</span></p><p>y = -x + 1</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the equation of the normal to the curve <span class="math-tex">\(y=e^{sin2x}\)</span> at <span class="math-tex">\((\frac{\pi}{2},1)\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(2x+4y-(4+\pi)=0\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(y=-e^2x+(1+e^2 \frac{\pi}{2})\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(2x-4y+(4-\pi)=0\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(y=-2x+(1+\pi)\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(y=e^{sin2x}\)</span></p><p><span class="math-tex">\(\frac{dy}{dx}=2cos2x\cdot e^{sin2x}\)</span></p><p>When x = <span class="math-tex">\(\frac{\pi}{2}\)</span>, <span class="math-tex">\(\frac{dy}{dx}=2cos(\pi)\cdot e^{sin(\pi)}\)</span></p><p><span class="math-tex">\(\frac{dy}{dx}=2(-1))\cdot e^{0}=-2\)</span></p><p>Gradient of tangent = -2</p><p>Gradient of normal = <span class="math-tex">\(\frac{-1}{-2}=\frac{1}{2}\)</span></p><hr class="hidden-separator"><p>Find the equation of the straight line with gradient = <span class="math-tex">\(\frac{1}{2}\)</span> which passes through the point <span class="math-tex">\((\frac{\pi}{2},1)\)</span></p><p><span class="math-tex">\(\frac{y-1}{x-\frac{\pi}{2}}=\frac{1}{2}\)</span></p><p><span class="math-tex">\(2y-2=x-\frac{\pi}{2}\)</span></p><p><span class="math-tex">\(4y-4=2x-\pi\)</span></p><p><span class="math-tex">\(2x-4y+(4-\pi)=0\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> The curve <span class="math-tex">\(y=\frac{lnx}{x}\)</span> has a maximum value at the point A.</p><p>Find the x coordinate of A</p></div><div class="q-answer"><p>x coordinate of A = <input type="text" style="height: auto;" data-c="e"> <span class="review"></span></p></div><div class="q-explanation"><p>Find the gradient function using the Quotient Rule</p><p><span></span><span class="math-tex">\(y=\frac { u }{ v } \\ \frac { dy }{ dx } =\frac { v\cdot \frac { du }{ dx } -u\cdot \frac { dv }{ dx } }{ v^{ 2 } } \)</span><span></span></p><hr class="hidden-separator"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>u = lnx</td><td>v = x</td></tr><tr><td><span class="math-tex">\(\frac{du}{dx}=\frac{1}{x}\)</span></td><td><span class="math-tex">\(\frac{dv}{dx}=1\)</span></td></tr></tbody></table><p><span></span> <span class="math-tex">\(\frac{dy}{dx}=\frac { x\cdot\frac{1}{x}-lnx\cdot 1 }{ x^{ 2 } } \)</span></p><p><span class="math-tex">\(\frac{dy}{dx}=\frac { 1-lnx }{ x^{ 2 } } \)</span></p><p>Maximum value occurs where <span class="math-tex">\(\frac{dy}{dx}=0 \)</span></p><p>Solve <span class="math-tex">\(\frac { 1-lnx }{ x^{ 2 } } =0\)</span></p><p>Since <span class="math-tex">\(x\neq 0\)</span> , 1-lnx = 0</p><p>lnx = 1</p><p>x = e</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> The function f(x) = tanx - 4x, <span class="math-tex">\(0 <x<\frac { \pi }{ 2 } \)</span> has a minimum value at <span class="math-tex">\(x = \frac { \pi }{ a} \)</span></p><p>Work out the value of <strong><em>a</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p>Differentiate the function f(x) = tanx - 4x</p><p><span class="math-tex">\(f'(x)=sec^2x-4\)</span></p><p>Minimum value occurs when <span class="math-tex">\(f'(x)=0\)</span></p><p>sec²x - 4 = 0</p><p>sec²x = 4</p><p>cos²x = <span class="math-tex">\(\frac{1}{4}\)</span></p><p>cosx = <span class="math-tex">\(\pm \frac { 1 }{ 2 } \)</span></p><p><span class="math-tex">\(Arccos(\frac{1}{2})=\frac{\pi}{3}\)</span></p><p>There is only one correct value in the interval <span class="math-tex">\(0 <x<\frac { \pi }{ 2 } \)</span></p><p>Hence <span class="math-tex">\(x=\frac{\pi}{3}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Find the x coordinate of the point of inflexion on the curve <span class="math-tex">\(y =xe^x\)</span></p></div><div class="q-answer"><p>x = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></p></div><div class="q-explanation"><p>Use the Product Rule to find <span class="math-tex">\(\frac{dy}{dx}\)</span></p><p><span class="math-tex">\(\frac{dy}{dx}=e^x+xe^x\)</span></p><p>Differentiate again to find <span class="math-tex">\(\frac{d²y}{dx²}\)</span></p><p><span class="math-tex">\(\frac{d²y}{dx²}=e^x+e^x+xe^x\)</span></p><p><span class="math-tex">\(\frac{d²y}{dx²}=e^x(2+x)\)</span></p><p>Point of inflexion exists where <span class="math-tex">\(\frac{d²y}{dx²}=0\)</span></p><p><span class="math-tex">\(e^x(2+x)=0\)</span></p><p><span class="math-tex">\(e^x=0\quad,\quad2+x=0\)</span></p><p>not possible , x = -2</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> <hr class="hidden-separator"> <p>Here's a quiz that practises <em><strong>Kinematics</strong></em></p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#bf5c5d23"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="bf5c5d23"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Mixed SL Differentiation 6 <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-391-2405" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>An object is moving along a line such that its displacement, <strong><em>s</em></strong> at any point in time <strong><em>t</em></strong> is given by</p><p>s(t) = 3sint - 4cost</p><p>What is the object's initial displacement?</p></div><div class="q-answer"><p>Initial displacement = <input type="text" style="height: auto;" data-c="-4"> <span class="review"></span></p></div><div class="q-explanation"><p>The initial displacement is given by s(0)</p><p>s(0) = 3sin(0) - 4cos(0)</p><p>s(0) = -4(1) = -4</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>An object moves in a straight line with position relative to the origin O given by</p><p>s(t) = sin4t - cos3t</p><p>where <strong><em>t</em></strong> is the time in seconds.</p><p>Which expression describes the velocity of the object?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{cos4t}{4} + \frac{sin3t}{3}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(4cos4t - 3sin3t\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(4cos4t + 3sin3t\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(-\frac{cos4t}{4} - \frac{sin3t}{3}\)</span></label></p></div><div class="q-explanation"><p>Velocity is given by the rate of change of displacement. We need to differentiate displacement:</p><p>s(t) = sin4t - cos3t</p><p><span class="math-tex">\(v(t)=s'(t) = 4cos4t + 3sin3t\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The velocity of a particle moving in a straight line is given by</p><p><span class="math-tex">\(v(t) = 10e^{-0.5t}\)</span></p><p>What is the initial acceleration of the particle?</p></div><div class="q-answer"><p>Initial acceleration = <input type="text" style="height: auto;" data-c="-5"> <span class="review"></span></p></div><div class="q-explanation"><p>Acceleration is the rate of change of velocity. We need to differentiate velocity</p><p><span class="math-tex">\(v(t) = 10e^{-0.5t}\)</span></p><p><span class="math-tex">\(a(t) = v'(t) = 10(-0.5)e^{-0.5t}\)</span></p><p><span class="math-tex">\(a(t) = -5e^{-0.5t}\)</span></p><p>Initial acceleration is a(0)</p><p><span class="math-tex">\(a(0) = -5e^{-0.5(0)}\)</span></p><p><span class="math-tex">\(a(0) = -5e^{0}=-5\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The displacement,<em><strong> s </strong></em>of an object is given by</p><p>s(t) = 1 - cos2t</p><p>where <em><strong>t </strong></em>is time.</p><p>The first time that the object's velocity is a maximum is when t = <span class="math-tex">\(\frac{\pi}{b}\)</span></p><p>Find <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p></div><div class="q-explanation"><p>Velocity is given by the rate of change of displacement. We need to differentiate displacement:</p><p>s(t) = 1 - cos2t</p><p><span class="math-tex">\(v(t)=s'(t) = 2sin2t\)</span></p><p>Maximum velocity occurs when <span class="math-tex">\(v'(t)=0\)</span></p><p><span class="math-tex">\(v'(t) = 4cos2t\)</span></p><p>Solve 4cos2t = 0</p><p>cos2t = 0</p><p>Arccos(0) = <span class="math-tex">\(\frac{\pi}{2}\)</span></p><p><span class="math-tex">\(2t=\frac{\pi}{2}\)</span></p><p><span class="math-tex">\(t=\frac{\pi}{4}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The displacement,<em><strong> s </strong></em>of an object is given by <span class="math-tex">\(s(t) = e^{-2t}\)</span>, where <em><strong>t </strong></em>is time.</p><p>Which is the correct expression for acceleration?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{e^{-2t}}{4}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(-2e^{-2t}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(4e^{-2t}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(-4e^{-2t}\)</span></label></p></div><div class="q-explanation"><p>Acceleration is the rate of change of velocity.</p><p>Velocity is given by the rate of change of displacement.</p><p>We need to differentiate two times:</p><p><span class="math-tex">\(s(t) = e^{-2t}\)</span></p><p><span class="math-tex">\(v(t)=s'(t) = -2e^{-2t}\)</span></p><p><span class="math-tex">\(a(t)=v'(t)=s''(t) = 4e^{-2t}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>A ball is thrown vertically upwards. The height, <strong><em>s </em></strong>of the ball after time <strong><em>t </em></strong> is given by the equation</p><p><span class="math-tex">\(s(t) = bt - 5t^2\)</span></p><p>If the ball reaches its maximum height when t = 2.5, find the value of <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="25"> <span class="review"></span></p></div><div class="q-explanation"><p>The ball reaches its maximum height when velocity = 0</p><p>To find velocity we need to differentiate the equation for displacement (height):</p><p><span class="math-tex">\(s(t) = bt - 5t^2\)</span></p><p><span class="math-tex">\(v(t) = s'(t) = b- 10t\)</span></p><p>v(t) = 0</p><p>b - 10t = 0</p><p>b = 10t</p><p>Since t = 2.5,</p><p>b= 10x2.5 = 25</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>An object is moving along a line such that its displacement, <strong><em>s</em></strong> at any point in time <strong><em>t</em></strong> is given by</p><p><span class="math-tex">\(s(t) = 2t^3-11t^2+16t-7\)</span></p><p>At what time does the object first change direction.</p></div><div class="q-answer"><p><strong><em>t</em></strong> = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"><p>The object changes direction when s(t) is a maximum or minimum</p><p><span class="math-tex">\(s(t) = 2t^3-11t^2+16t-7\)</span></p><p><span class="math-tex">\(s'(t) = 6t^2-22t+16\)</span></p><p>Solve <span class="math-tex">\(s'(t) = 0\)</span></p><p><span class="math-tex">\(6t^2-22t+16=0\\
3t^2-11t+8=0\)</span></p><p>(3t - 8)(t - 1) = 0</p><p><span class="math-tex">\(t = \frac{8}{3}\quad,\quad t=1\)</span></p><p>Let's check that s(1) is a maximum or minimum</p><p><span class="math-tex">\(s'(t) = 6t^2-22t+16\)</span></p><p><span class="math-tex">\(s''(t) = 12t-22\)</span></p><p><span class="math-tex">\(s''(1) = 12-22<0\)</span> Maximum</p><p>The object first changes direction when t = 1</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>An object is moving along a line such that its displacement, <strong><em>s</em></strong> at any point in time <strong><em>t</em></strong> is given by</p><p><span class="math-tex">\(s(t) = sint+cost\)</span></p><p>What is the minimum velocity of the object?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 0</label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\sqrt{2}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{3\pi}{4}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(-\sqrt{2}\)</span></label></p></div><div class="q-explanation"><p>The minimum velocity occurs when acceleration = 0</p><p><span class="math-tex">\(s(t) = sint+cost\)</span></p><p><span class="math-tex">\(v(t)=s'(t) = cost-sint\)</span></p><p><span class="math-tex">\(a(t)=v'(t)=s''(t) = -sint-cost\)</span></p><p>a(t) = 0</p><p>-sint - cost = 0</p><p>sint = -cost</p><p><span class="math-tex">\(\frac{sint}{cost}=-1\)</span></p><p>tant = -1</p><p><span class="math-tex">\(t = \frac{3\pi}{4}\)</span></p><p>Minimum velocity <span class="math-tex">\(v( \frac{3\pi}{4}) = cos( \frac{3\pi}{4})-sin( \frac{3\pi}{4})\)</span></p><p><span class="math-tex">\(v( \frac{3\pi}{4}) = -\frac{1}{\sqrt{2}} -\frac{1}{\sqrt{2}}\)</span></p><p><span class="math-tex">\(=-\frac{2}{\sqrt{2}}\\=-\sqrt{2}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <p> </p> <div class="panel panel-green panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test yourself further</p> </div> </div> <div class="panel-body"> <div> <p>You can get further practice by trying a dynamic quiz.</p> <p>If you refresh this page, you will get a new set of quizzes</p> <div class="panel panel-green panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> Easy</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz dyn-quiz" data-pid="2405" data-mtime="1675506255.4496"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p><span class="math-tex">\(\frac{d}{dx}\frac{2}{(3x-5)^{4}} \)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{-24}{(3x-5)^{3}}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac{-24}{(3x-5)^{5}}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{-8}{(3x-5)^{5}}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{-8}{(3x-5)^{3}}\)</span></label></p></div><div class="q-explanation"><p>It might help to write question as <span class="math-tex">\(\frac{d}{dx}{2}{(3x-5)^{-4}} \)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(f(x)\ =\ \frac { 1 }{ x } -\frac { 3 }{ { x }^{ 2 } } \)</span> then <span class="math-tex">\(f'(x)=\frac { -1 }{ { x }^{ a } } +\frac { b }{ { x }^{ c } } \)</span></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="6"> <span class="review"></span></p><p>c = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(f(x)=\frac { 1 }{ x } -\frac { 3 }{ { x }^{ 2 } } ={ x }^{ -1 }-3{ x }^{ -2 }\\ \Rightarrow f'(x)=-1{ x }^{ -2 }-3{ \cdot (-2)x }^{ -3 }=\frac { -1 }{ { x }^{ 2 } } +\frac { 6 }{ { x }^{ 3 } } \)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>The following diagram show a graph of <strong><em>f' </em></strong>, the derivative of <strong><em>f</em></strong></p><p><strong><em><img alt="" src="../../files/differentiation/graphs/q2.jpg" style="width: 300px; height: 236px;"></em></strong></p></div><div class="q-answer"><p>How many stationary points does <strong><em>f </em></strong> have? <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p>Stationary points when <span class="math-tex">\(f' (x) = 0\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>Differentiate sin2x</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\({cos2x \over 2}\)</span></label></p><p><label class="radio"><input type="radio"> -2cos2x</label></p><p><label class="radio"><input class="c" type="radio"> 2cos2x</label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(-{cos2x \over 2}\)</span></label></p></div><div class="q-explanation"><p>You should try to remember that <span class="math-tex">\(\frac { d }{ dx } (sinax)=acosax\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out <span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(\ln 2x)\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large2x\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large\frac{1}{2x}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large\frac{1}{x}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large\frac{2}{x}\)</span></label></p></div><div class="q-explanation"><p>Using the laws of logarithms, we know that</p><p><span class="math-tex">\(\large\ln 2x=\ln 2+\ln x\)</span></p><p><span class="math-tex">\(\large\frac{\mathrm{d}}{\mathrm{d}x}(\ln 2+\ln x)=0+\frac{1}{x}\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-green panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> Medium</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz dyn-quiz" data-pid="2405" data-mtime="1675506255.4532"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Differentiate <span class="math-tex">\(x^{2}lnx\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 2xlnx</label></p><p><label class="radio"><input class="c" type="radio"> x + 2xlnx</label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(2xlnx+{ 1 \over x}\)</span></label></p><p><label class="radio"><input type="radio"> 2xlnx + x²e<sup>x</sup></label></p></div><div class="q-explanation"><p>We need to use the product rule</p><p><span class="math-tex">\(y=uv\\ \frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}\)</span></p><hr class="hidden-separator"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>u = x²</td><td>v = lnx</td></tr><tr><td><span class="math-tex">\(\frac { du }{ dx } \)</span> = 2x</td><td><span class="math-tex">\(\frac { dv }{ dx } ={1 \over x}\)</span></td></tr></tbody></table><p><span class="math-tex">\( \frac{dy}{dx}=x^2 \times{1 \over x} \ +lnx\times 2x\)</span></p><p>= x + 2xlnx</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Differentiate <span class="math-tex">\(\frac{x^2+1}{3x-2}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { -3x^2+4x +1}{ (3x-2)^{ 2 } } \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { 3x^2-4x +3}{ (3x-2)^{ 2 } } \)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac { 3x^2-4x -3}{ (3x-2)^{ 2 } } \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { -3x^2-4x +1}{ (3x-2)^{ 2 } } \)</span></label></p></div><div class="q-explanation"><p>We need to use the quotient rule</p><p><span class="math-tex">\(y=\frac { u }{ v } \\ \frac { dy }{ dx } =\frac { v\cdot \frac { du }{ dx } -u\cdot \frac { dv }{ dx } }{ v^{ 2 } } \)</span></p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>u = x² + 1</td><td>v = 3x - 2</td></tr><tr><td><span class="math-tex">\(\frac{du}{dx}=2x\)</span></td><td><span class="math-tex">\(\frac{dv}{dx}=3\)</span></td></tr></tbody></table><p> <span class="math-tex">\(\frac { dy }{ dx } =\frac { (3x-2)\cdot (2x) -(x^2+1)\cdot 3}{ (3x-2)^{ 2 } } \)</span></p><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { 6x^2-4x -3x^2-3}{ (3x-2)^{ 2 } } \)</span></p><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { 3x^2-4x -3}{ (3x-2)^{ 2 } } \)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>Differentiate <span class="math-tex">\(\frac { ln3x }{ { e }^{ 2x } } \)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { { e }^{ 2x }-2x{ e }^{ 2x }ln3x }{ { e }^{ 2x } } \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { { e }^{ 2x }-2x{ e }^{ 2x }ln3x }{ { xe }^{ 4x^{ 2 } } } \)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac { 1-2xln3x }{ { xe }^{ 2x } }\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { \frac { { e }^{ 2x } }{ x } -2{ e }^{ 2x }ln3x }{ { e }^{ 4x^{ 2 } } } \)</span></label></p></div><div class="q-explanation"><p>We need to use the Quotient Rule</p><p><span class="math-tex">\(y=\frac { u }{ v } \\ \frac { dy }{ dx } =\frac { v\cdot \frac { du }{ dx } -u\cdot \frac { dv }{ dx } }{ v^{ 2 } } \)</span></p><hr class="hidden-separator"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>u = ln3x</td><td>v = <span class="math-tex">\(e^{2x}\)</span></td></tr><tr><td><span class="math-tex">\(\frac { du }{ dx } =\frac{1}{x}\)</span></td><td><p><span class="math-tex">\(\frac { dv }{ dx } =2e^{2x}\)</span></p></td></tr></tbody></table><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { e^{ 2x }\cdot \frac { 1 }{ x } -2e^{ 2x }\cdot ln3x }{ ({ e }^{ 2x })^{ 2 } } \)</span></p><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { e^{ 2x }(\frac { 1 }{ x } -2ln3x) }{ ({ e }^{ 2x })^{ 2 } } \)</span></p><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { \frac { 1 }{ x } -2ln3x }{ { e }^{ 2x } } \)</span></p><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { \frac { 1 }{ x } -\frac { 2xln3x }{ x } }{ { e }^{ 2x } } \)</span></p><p><span class="math-tex">\(\frac { dy }{ dx } =\frac { 1-2xln3x }{ { xe }^{ 2x } } \)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> The function f(x) = tanx - 4x, <span class="math-tex">\(0 <x<\frac { \pi }{ 2 } \)</span> has a minimum value at <span class="math-tex">\(x = \frac { \pi }{ a} \)</span></p><p>Work out the value of <strong><em>a</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p>Differentiate the function f(x) = tanx - 4x</p><p><span class="math-tex">\(f'(x)=sec^2x-4\)</span></p><p>Minimum value occurs when <span class="math-tex">\(f'(x)=0\)</span></p><p>sec²x - 4 = 0</p><p>sec²x = 4</p><p>cos²x = <span class="math-tex">\(\frac{1}{4}\)</span></p><p>cosx = <span class="math-tex">\(\pm \frac { 1 }{ 2 } \)</span></p><p><span class="math-tex">\(Arccos(\frac{1}{2})=\frac{\pi}{3}\)</span></p><p>There is only one correct value in the interval <span class="math-tex">\(0 <x<\frac { \pi }{ 2 } \)</span></p><p>Hence <span class="math-tex">\(x=\frac{\pi}{3}\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>The velocity of a particle moving in a straight line is given by</p><p><span class="math-tex">\(v(t) = 10e^{-0.5t}\)</span></p><p>What is the initial acceleration of the particle?</p></div><div class="q-answer"><p>Initial acceleration = <input type="text" style="height: auto;" data-c="-5"> <span class="review"></span></p></div><div class="q-explanation"><p>Acceleration is the rate of change of velocity. We need to differentiate velocity</p><p><span class="math-tex">\(v(t) = 10e^{-0.5t}\)</span></p><p><span class="math-tex">\(a(t) = v'(t) = 10(-0.5)e^{-0.5t}\)</span></p><p><span class="math-tex">\(a(t) = -5e^{-0.5t}\)</span></p><p>Initial acceleration is a(0)</p><p><span class="math-tex">\(a(0) = -5e^{-0.5(0)}\)</span></p><p><span class="math-tex">\(a(0) = -5e^{0}=-5\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-green panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> Hard</p> </div> </div> <div class="panel-body"> <div class="tib-quiz dyn-quiz" data-pid="2405" data-mtime="1675506255.4561"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>The following is true about a function <strong><em>f</em></strong></p><ul><li><span class="math-tex">\(a<b<c\)</span></li><li>f(x) is increasing on the interval [a,c]</li><li><span class="math-tex">\(f'(x)<0\)</span> for <span class="math-tex">\(x<a\)</span></li><li><span class="math-tex">\(f''(b)=0\)</span></li><li><span class="math-tex">\(f'(c)=0\)</span><hr class="hidden-separator"></li></ul><p>Which of the following is TRUE?</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(f'(a)=0\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> There is a local minima at x = a</label></p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(f'(b)=0\)</span></label><p><label class="checkbox"><input class="c" type="checkbox"> There is a point of inflexion at x = b</label></p></div><div class="q-explanation"><p><img alt="" src="../../files/differentiation/mixed-differentiation/quiz4-graphs/q10explanation.png" style="width: 500px; height: 452px;"></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Find the equation of the normal to the curve <span class="math-tex">\(y=x^3lnx\)</span> at (1,0)</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> y = 4x - 4</label></p><p><label class="radio"><input type="radio"> y = x - 1</label></p><p><label class="radio"><input class="c" type="radio"> y = -x + 1</label></p><p><label class="radio"><input type="radio"> 4y = -x + 1</label></p></div><div class="q-explanation"><p>We need to use the product rule to differentiate <span class="math-tex">\(y=x^3lnx\)</span></p><p><span></span><span class="math-tex">\(y=uv\\ \frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}\)</span><span></span></p><hr class="hidden-separator"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>u = <span class="math-tex">\(x^3\)</span></td><td>v = lnx</td></tr><tr><td><span class="math-tex">\(\frac{du}{dx}=3x^2\)</span></td><td><span class="math-tex">\(\frac{dv}{dx}=\frac{1}{x}\)</span></td></tr></tbody></table><p><span></span> <span class="math-tex">\(\frac{dy}{dx}=x^3\cdot \frac{1}{x}+lnx\cdot3x^2\)</span></p><p><span class="math-tex">\(\frac{dy}{dx}=x^2+3x^2lnx\)</span></p><p><span class="math-tex">\(\frac{dy}{dx}=x^2(1+lnx)\)</span></p><p>When x = 1, <span class="math-tex">\(\frac{dy}{dx}=1(1+ln1)=1(1+0)=1\)</span></p><p>Gradient of tangent = 1</p><p>Gradient of normal = -1</p><hr class="hidden-separator"><p>Find the equation of the straight line with gradient = -1 which passes through the point (1,0)</p><p><span class="math-tex">\(\frac{y-0}{x-1}=-1\)</span></p><p>y = -x + 1</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>The following is true about a function <strong><em>f</em></strong></p><ul><li><span class="math-tex">\(a<><c\)< span=""><><><><c< li=""> <><><c<><c\)<></c\)<></c<></c<></c\)<></span></li><li>f(x) is increasing on the interval [a,c]</li><li><span class="math-tex">\(f'(x)<0\)</span> for <span class="math-tex">\(x<a\)< span=""><><a< li=""> <a<></a<></a<></a\)<></span></li><li><span class="math-tex">\(f''(b)=0\)</span></li><li><span class="math-tex">\(f'(c)=0\)</span><hr class="hidden-separator"></li></ul><p>Which of the following is TRUE?</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> There is a point of inflexion at x = b</label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(f'(a)=0\)</span></label></p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(f'(b)=0\)</span></label><p><label class="checkbox"><input class="c" type="checkbox"> There is a local minima at x = a</label></p></div><div class="q-explanation"><p><img alt="" src="../../files/differentiation/mixed-differentiation/quiz4-graphs/q10explanation.png" style="width: 500px; height: 452px;"></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> Use your graphic display calculator to solve this question</p><p>Let <span class="math-tex">\(\large f(x)=sin(e^x) \ , \ 0\le x \le 2\)</span></p><p>The curve of <em><strong>f</strong></em> is concave up on the interval<strong><em> a < x < b</em></strong></p><p>Find the values of <strong><em>a </em></strong>and <strong><em>b</em></strong></p><p><strong><em>Give your answer to 3 s.f. </em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="1.23"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="1.86"> <span class="review"></span></p></div><div class="q-explanation"><p>The curve of a function is concave up when <span class="math-tex">\(\large f''(x)>0\)</span></p><p>This will occur between two points of inflexion</p><p><img alt="" src="../../files/differentiation/using-gdc/q9ans.png" style="width: 400px; height: 361px;"></p><p>Differentiate <span class="math-tex">\(\large f(x)=sin(e^x) \)</span> using the Chain Rule</p><p><span class="math-tex">\(\large f'(x)=e^xcos(e^x) \)</span></p><p>We can find the points of inflexion for <span class="math-tex">\(f(x)\)</span> by finding the stationary points for <span class="math-tex">\(f'(x)\)</span></p><p><img alt="" src="../../files/differentiation/using-gdc/q9ans2.png" style="width: 681px; height: 177px;"></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> Use your graphic display calculator to solve this question</p><p>Let <span class="math-tex">\(\large f(x)=2x \ln x \ ,\ x>0\)</span></p><p>Points <em><strong>P</strong></em>(1,0) and <em><strong>Q</strong></em> are on the curve of <strong><em>f</em></strong>.</p><p>The tangent to the curve of <strong><em>f</em></strong> at <em><strong>P</strong></em> is perpendicular to the tangent to the curve at <em><strong>Q</strong></em>.</p><p>Find the x coordinates of <em><strong>Q</strong></em>.</p> <p><strong><em>Give your answer to 3 s.f. </em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="0.287"> <span class="review"></span></p></div><div class="q-explanation"><p>You do not need to plot a graph of the curve and the tangents, but it may help you visualise the problem.</p><p>Plot the graph of the function and the tangent to the curve at the point x = 1</p><p>To get a true picture of the graph, you may want to plot the graph so that the axes are 1:1 (so that a right angle actually looks like a right angle).</p><p><img alt="" src="../../files/differentiation/using-gdc/q10ans1.png" style="width: 268px; height: 154px;"></p><p>You will notice that the gradient of this tangent = 2</p><p>You need to find the x coordinates of the graph for which the curve has a gradient = <span class="math-tex">\(-\frac{1}{2}\)</span></p><p>Differentiate the function using the product rule</p><p><span class="math-tex">\(\large f'(x)=2 \ln x+2 \)</span></p><p>Solve <span class="math-tex">\(\large 2 \ln x+2 =-\frac{1}{2}\)</span></p><p>You can use the equation solver or a graphical approach.</p><p>Here is what the two tangents look like</p><p><img alt="" src="../../files/differentiation/using-gdc/q10ans2.png" style="width: 268px; height: 154px;"></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="2405"> <div class="panel-heading">MY PROGRESS</div> <div class="panel-body understanding-rate"> <div class="msg"></div> <label class="label-lg">Self-assessment</label><p>How much of <strong>HL Mixed Differentiation</strong> have you understood?</p><div class="slider-container text-center"><div id="self-assessment-slider" class="sib-slider self-assessment " data-value="1" data-percentage=""></div></div> <label class="label-lg">My notes</label> <textarea name="page-notes" class="form-control" rows="3" placeholder="Write your notes here..."></textarea> </div> <div class="panel-footer text-xs-center"> <span id="last-edited" class="mb-xs-3"> </span> <div class="actions mt-xs-3"> <button id="save-my-progress" type="button" class="btn btn-sm btn-primary text-center btn-xs-block"> <i class="fa fa-fw fa-floppy-o"></i> Save </button> </div> </div></div> <div id="modal-feedback" class="modal fade" tabindex="-1" role="dialog"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <h4 class="modal-title">Feedback</h4> <button type="button" class="close hidden-xs hidden-sm" data-dismiss="modal" aria-label="Close"> <span aria-hidden="true">×</span> </button> </div> <div class="modal-body"> <div class="errors"></div> <p><strong>Which of the following best describes your feedback?</strong></p> <form method="post" style="overflow: hidden"> <div class="form-group"> <div class="radio"><label style="color: #121212;"><input type="radio" name="feedback-type" value="Recommendation"> Recommend</label></div><div class="radio"><label style="color: #121212;"><input type="radio" name="feedback-type" value="Problem"> Report a problem</label></div><div class="radio"><label style="color: #121212;"><input type="radio" name="feedback-type" value="Improvement"> Suggest an improvement</label></div><div class="radio"><label style="color: #121212;"><input type="radio" name="feedback-type" value="Other"> Other</label></div> </div> <hr> <div class="row"> <div class="col-md-6"> <div class="form-group"> <label for="feedback-name">Name</label> <input type="text" class="form-control" name="feedback-name" placeholder="Name" value=" "> </div> </div> <div class="col-md-6"> <div class="form-group"> <label for="feedback-email">Email address</label> <input type="email" class="form-control" name="feedback-email" placeholder="Email" value="@airmail.cc"> </div> </div> </div> <div class="form-group"> <label for="feedback-comments">Comments</label> <textarea class="form-control" name="feedback-comments" style="resize: vertical;"></textarea> </div> <input type="hidden" name="feedback-ticket" value="082b9c9c4ae3624d"> <input type="hidden" name="feedback-url" value="https://studyib.net/mathsanalysis/page/2405/hl-mixed-differentiation"> <input type="hidden" name="feedback-subject" value="11"> <input type="hidden" name="feedback-subject-name" value="Maths: Analysis & Approaches"> <div class="pull-left"> </div> </form> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary btn-xs-block feedback-submit mb-xs-3 pull-right"> <i class="fa fa-send"></i> Send </button> <button type="button" class="btn btn-default btn-xs-block m-xs-0 pull-left" data-dismiss="modal"> Close </button> </div> </div> </div></div> </article> <hr class="hidden-md hidden-lg"> <div class="hidden-md hidden-lg mt-xs-3"> <button class="btn btn-default btn-block text-xs-center" data-toggle="modal" data-target="#modal-feedback" style="margin-bottom: 10px"><i class="fa fa-send"></i> Feedback</button> </div> </div> <input type="hidden" id="user-id" value="38342"></div><input id="ticket" type="hidden" value="082b9c9c4ae3624d"><input id="tzoffset" type="hidden" value="new"><input id="fp" class="fp" type="hidden" value=""></div><div id="std-footer"> <div class="wmap"> <div class="layout-wrapper"> <p> <a href="https://www.inthinking.net"> © <span id="footer-year"></span> <em>InThinking</em> <script>document.getElementById("footer-year").innerHTML = new Date().getFullYear();</script> </a> | <a target="_self" href="../../../about.html"> About us </a> | <a target="_self" href="../../../terms-and-conditions.html"> Legal </a> | <a target="_self" href="../../../contact.html"> Contact </a> </p> <p> <a class="social" target="_blank" href="https://twitter.com/#!/inthinker"> <img src="../../../img/social/twitter-square.svg"> Twitter </a> <a class="social" target="_blank" href="https://www.facebook.com/inthinking.net"> <img src="../../../img/social/facebook-square.svg"> Facebook </a> <a class="social" target="_blank" href="https://www.linkedin.com/profile/view?id=139741989"> <img src="../../../img/social/linkedin-square.svg"> LinkedIn </a> </p> </div> </div></div><script src="../../../js/jquery-1.11.3.min.js"></script><script src="../../../js/bootstrap.min.js"></script><script src="../../../js/jquery-fancybox/jquery.fancybox.min.js"></script><script>var pageId = 2405;</script><script type="text/javascript" src="../../../js/jqueryui/jquery-ui-custom.min.js"></script><script type="text/javascript" src="../../../js/jqueryui/jquery.ui.touch-punch.js"></script><script type="text/javascript" src="../../../js/std-quizzes-helpers.min.js?v=20220406"></script><script type="text/javascript" src="../../../js/std-quizzes.min.js?v=2022041300"></script><script src="../../../ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script><script type="text/javascript" src="../../../js/slick-carousel/slick.min.js"></script><script type="text/javascript" src="../../../js/std-slide-quizzes.min.js?v=20220406"></script><script type="text/javascript" src="../../../js/jqueryui/jquery-ui.min.js"></script><script type="text/javascript" src="../../../js/jqueryui/jquery.ui.touch-punch.min.js"></script><script src="../../../js/user/page-my-progress.min.js?v=202211221000"></script><script>var sAJAX='/pages/subjects/activity/user-stats-page.php?t=082b9c9c4ae3624d&p=2405&s=11&x=248';var sData='k4iA6GSB03BgPm14adQLIxiA6GSB0bfq7Y9FwFiA6GSBPFAF7YcHEKvrZ0fREzP5PgsOPuLjkYKgPufjEnEB7nK2SYBsSYBhWFPRPHpsE3KeSYcH74P5k4irWbpj6bOFwBRbOUSSIxiDZ3BRWVOFwBR3acrRPnyBSnKR74P5GDaSezAFWGcrE0ij6GyDPmjn6GyDEzAF7bKFEYv86GB2Pm1FPFAF7YcrZxP5PBAhWGcrZVasWncRk0aj7LAh7YcHEKAhOmCAaKAhZYA8WGBNEGC8EYBnEnK4EGqrZGcrZGv2PHr=';var loopSecs = 30;var minSecsLog = 60;var lsKey = '11-83608-2405';</script><script src="../../../js/subjects/activity/user-stats-page.min.js?v=202205311700"></script><script src="../../../js/subjects/activity/my-favorites.min.js?v=20220610"></script><script src="../../../js/subjects/feedback.min.js?v=202211221000"></script><script>var sessionUpdateSecs = 300;</script><script type="text/javascript" src="../../../js/session-updater.js"></script><script>(function(i,s,o,g,r,a,m){i['GoogleAnalyticsObject']=r;i[r]=i[r]||function(){(i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o),m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m)})(window,document,'script','https://www.google-analytics.com/analytics.js','ga');ga('create', 'UA-98480796-4', 'auto');ga('send', 'pageview');</script><script src="../../../js/devicefp/devicefp.min.js"></script><script src="../../../js/jq-quickfit/jq-quickfit.min.js?v=20220201"></script><script src="../../../js/frontpage-subjects.min.js?v=202211161300"></script><script type="text/javascript" src="../../../js/studyib-jq.min.js?v=202211241300"></script><script type="text/javascript">$(document).ready(function(){ $('#trigger-modal-video-overview').click(function() { if( $('#modal-video-overview .modal-body iframe').attr('src').length == 0 ) { $('#modal-video-overview .modal-body iframe').attr('src', 'https://www.youtube.com/embed/hxxtdiLdTFk?rel=0'); } }); $('#modal-video-overview').on('hidden.bs.modal', function() { $('#modal-video-overview .modal-body iframe').attr('src', ''); }); });</script></body></html>