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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"> Chain Rule <a href="#" class="mark-page-favorite pull-right" data-pid="742" 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">Chain Rule</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="../../chain-rule-1.jpg" style="float: left; width: 100px; height: 100px;">The Chain Rule is used for differentiating composite functions. The rule itself looks really quite simple (and it is not too difficult to use). The most important thing to understand is when to use it and then get lots of practice. It is useful to get fluent in applying The Chain Rule, as this will save you lots of time in an exam. This is especially true since some functions are composed of more than two functions and The Chain Rule is often used as part of a bigger question that use other rules ( <a href="../743/product-and-quotient-rule.html" title="Product and Quotient Rule">Product and Quotient Rule</a> ) or questions about Related Rates of Change or <a href="../741/implicit-differentiation.html" title="Implicit Differentiation">Implicit Differentiation</a>.</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 should learn about</p> <ul> <li>differentiating composite functions using the chain rule</li> </ul> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <p>The following videos will help you understand all the concepts from this page</p> <div class="panel panel-yellow panel-has-colored-body panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Why and How</p> </div> </div> <div class="panel-body"> <p>The following video explains why we need The Chain Rule and how to use it. There will be two examples in which we will find the derivative of <span class="math-tex">\((3x+1)^4\)</span> and <span class="math-tex">\(e^{3x^2}\)</span></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/219191217"></iframe></div> </div> </div> <div class="panel panel-yellow 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>Differentiating [f(x)]<sup>n</sup></p> </div> </div> <div class="panel-body"> <div> <div> <p>In the following video, we will look for some simple patterns so that we can quickly differentiate functions in the form [f(x)]<sup>n</sup></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/219192677"></iframe></div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-expandable panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Differentiating e<sup>f(x)</sup> and ln[f(x)]</p> </div> </div> <div class="panel-body"> <div> <p>In the following video, we will look for some simple patterns so that we can quickly differentiate functions in the form <span calibri=""><em>e<sup>f(x)</sup> and </em><span calibri=""><em>ln[f(x)]</em></span></span></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/220176760"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <p><iframe align="middle" frameborder="1" height="480" scrolling="yes" src="../../quick-rule-cheat-sheet.pdf" width="640"></iframe></p> <p>Print from <a href="../../quick-rule-cheat-sheet.pdf" target="_blank">here</a></p> </div> </div> <div class="panel-footer"> <div> <p>text</p> </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>We have seen that <span class="math-tex">\(\frac{d}{dx}[f(x)]^{n}=n\cdot f\,'(x)\cdot[f(x)]^{n-1}\)</span></p> <p>Here is a quiz that will get you to practise deriving simple expressions using this quick rule</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#1ecee13d"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="1ecee13d"> <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%;"> Chain Rule - quick rule f(x)^n <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="11-157-742" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p><span class="math-tex">\(\frac{d}{dx}(3x-4)^{6}=\boldsymbol{a}(3x-4)^{\boldsymbol{b}}\)</span></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="18"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p></div><div class="q-explanation"></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">\(\frac{d}{dx}(2-4x)^{3}=\boldsymbol{a}(2-4x)^{\boldsymbol{b}}\)</span></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="-12"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"></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">\(\frac{d}{dx}(\frac{1}{(lnx)^{2}})=\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{-1}{x(lnx)^{3}}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{2}{x}(lnx)^{3}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{-2}{x(lnx)}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac{-2}{x(lnx)^{3}}\)</span></label></p></div><div class="q-explanation"></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">\(\frac{d}{dx}(cos^{3}x)=\frac{d}{dx}(cosx)^{3}=\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(3sinx(cos^{2}x)\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(3cos^{2}x\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(-3sinx(cos^{2}x)\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(3sin^{2}x\)</span></label></p></div><div class="q-explanation"></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>We have also seen that</p> <ul> <li><span class="math-tex">\(\frac{d}{dx}e^{f(x)}=f\,'(x)e^{f(x)}\)</span></li> <li><span class="math-tex">\(\frac{d}{dx}ln[f(x)] = \frac{f\,'(x)}{f(x)}\)</span></li> </ul> <p>Here is a quiz that will get you to practise deriving simple expressions using these quick rules</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#d6dc28ba"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="d6dc28ba"> <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%;"> Chain Rule - quick rule e^f(x), ln(f(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="11-159-742" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><span class="math-tex">\(\frac{d}{dx}(e^{x^{3}-2x})=\)</span></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\((3x^2-2)e^{x^{3}-2x}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\((x^{3}-2x)e^{x^{3}-2x}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\((x^{3}-2x)e^{3x^{2}-2}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(3x^{2}e^{x^{3}-2x}\)</span></label></p></div><div class="q-explanation"><p>Hidden explanation (optional)</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">\(\frac{d}{dx}e^{cos2x}=\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(sin2xe^{cos2x}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(2sin2xe^{cos2x}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(-sinxe^{cos2x}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(-2sin2xe^{cos2x}\)</span></label></p></div><div class="q-explanation"></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">\(\frac{d}{dx}e^{sin^{2}x}=\)</span></p><p><em>Can you find 2 correct answers?</em></p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(sin2xe^{sin^{2}x}\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(cosxe^{sinx}\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(2cosxsinxe^{sin^{2}x}\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(2cosxe^{sinx}\)</span></label></p></div><div class="q-explanation"><p>Note that 2 answers are correct since <span class="math-tex">\(sin2x\equiv2sinxcosx\)</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">\(\frac{d}{dx}ln(2x+5)= \)</span></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac{2}{2x+5}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{2x+5}{2}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{5}{2x+5}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{1}{2x+5}\)</span></label></p></div><div class="q-explanation"></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">\(\frac{d}{dx}ln(sin2x) = \)</span></p><p><em>Can you find 2 correct answers?</em></p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(\frac{2cos2x}{sin2x}\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(\frac{sin2x}{2cos2x}\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(\frac{cos2x}{sin2x}\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(2\cot2x\)</span></label></p></div><div class="q-explanation"><p>2 answers are correct since cotx = <span class="math-tex">\(\frac{1}{tanx}=\frac{1}{\frac{sinx}{cosx}}=\frac{cosx}{sinx}\)</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 class="exercise shadow-bottom"><div class="q-question"><p><span class="math-tex">\(\frac{d}{dx}ln(cos^{2}x)=\)</span></p><p><em>Can you find 2 correct answers?</em></p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(\frac{sinx}{cos^{2}x}\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(\frac{2cosx}{cos^{2}x}\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> -2tanx</label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(\frac{-sin2x}{cos^{2}x}\)</span></label></p></div><div class="q-explanation"><p>2 correct answers</p><p><span class="math-tex">\(\frac{d}{dx}ln(cos^{2}x)=\frac{d}{dx}ln(cosx)^2=\frac{-2sinxcosx}{cos^{2}x}=\frac{-2sinx}{cosx}=-2tanx\)</span></p><p>or</p><p><span class="math-tex">\(\frac{d}{dx}ln(cos^{2}x)=\frac{d}{dx}ln(cosx)^2=\frac{-2sinxcosx}{cos^{2}x}=\frac{-sin2x}{cos^2x}\)</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> </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 is a mixed quiz that will get you to practise <strong>all the skills</strong> deriving simple expressions using the Chain Rule</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#01a3bd9a"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="01a3bd9a"> <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%;"> Chain Rule - mixed <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="11-160-742" style="opacity: 0"> <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">\(\frac{-8}{(3x-5)^{3}}\)</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">\(\frac{-24}{(3x-5)^{3}}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{-8}{(3x-5)^{5}}\)</span></label></p></div><div class="q-explanation"><p>It might help to write the question as <span class="math-tex">\(\frac{d}{dx}{2}{(3x-5)^{-4}} \)</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><em><strong>f</strong>(x)</em> = <span class="math-tex">\(e^{3x}\)</span></p><p>Find <em>f<strong>'</strong>(ln2)</em></p></div><div class="q-answer"><p><em>f</em><strong><em>'</em></strong><em>(ln2)</em><strong><em> = </em></strong><input type="text" style="height: auto;" data-c="24"> <span class="review"></span></p></div><div class="q-explanation"><p><em><strong>f'</strong>(x)</em> = 3<span class="math-tex">\(e^{3x}\)</span></p><p><em>f</em><strong><em>'</em></strong><em>(ln2)</em><strong><em> =</em></strong>3<span class="math-tex">\(e^{3ln2}\)</span> = 3<span class="math-tex">\(e^{ln2^{3}}\)</span> = 3<span class="math-tex">\(e^{ln8}\)</span> = 3<span class="math-tex">\(\times\)</span>8 = 24</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">\(y = \ln(\tan x)\)</span></p><p>Find <span class="math-tex">\(\frac{d y}{dx} \)</span></p><p>There is more than one correct answer</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(\sec x \mathrm{cosec} x \)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(\frac{\sin x}{\cos x}\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(\frac{\sec x}{\tan x}\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(\frac{\sec^{2}x}{\tan x}\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(y = \ln(\tan x)\\ y = \ln u \qquad u=\tan x\\ \frac{d y}{du}= \frac{1}{u} \qquad\frac{du}{dx}= \sec^{2}x \\ \frac{dy}{dx}= \frac{\sec^{2}x}{\tan x}=\frac{1}{\cos^{2}x}\times\frac{\cos x}{\sin x}=\frac{1}{\cos x\sin x}=\sec x\mathrm{cosec}x\)</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 the gradient to the curve y = tan(ln(x)) when 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>y = tan(ln(x))</p><p><span class="math-tex">\(y = tan u \qquad u = lnx \\ \frac{dy}{du}= sec^{2}u \quad\frac{du}{dx}= \frac{1}{x}\)</span> </p><p><span class="math-tex">\(\frac{dy}{dx}= \frac{sec^{2}(lnx)}{x}\)</span></p><p>when x = 1, <span class="math-tex">\(\frac{dy}{dx}= \frac{sec^{2}(ln1)}{1}\)</span></p><p>ln1 = 0 , sec²0 =</p><p><span class="math-tex">\(\frac{dy}{dx}=1\)</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> y = <span class="math-tex">\(e^{cos^{2}2x}\)</span></p><p> When x = <span class="math-tex">\( \frac{\pi}{6}\)</span>, <span class="math-tex">\( \frac{dy}{dx}\)</span> can be expressed in the form <span class="math-tex">\( -\sqrt{\boldsymbol{a}}e^{0.25}\)</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="3"> <span class="review"></span></p></div><div class="q-explanation"><p> y = <span class="math-tex">\(e^{cos^{2}2x}\)</span></p><p>y = <span class="math-tex">\(e^{u}\)</span> <span class="math-tex">\(u = (cos2x)^2\)</span></p><p><span class="math-tex">\( \frac{dy}{du}=e^{u}\)</span> <span class="math-tex">\( \frac{du}{dx}=2(-2sin2x)(cos2x)^{1}\)</span></p><p><span class="math-tex">\( \frac{dy}{dx}= \frac{dy}{du}\times \frac{du}{dx}\)</span></p><p><span class="math-tex">\( \frac{dy}{dx}= e^{u}\times (-4sin2xcos2x)\)</span></p><p><span class="math-tex">\( \frac{dy}{dx}=-4sin2xcos2x e^{(cos^{2}2x)}\)</span></p><p> When x = <span class="math-tex">\( \frac{\pi}{6}\)</span>, <span class="math-tex">\( \frac{dy}{dx}=-4sin2 (\frac{\pi}{6})cos2( \frac{\pi}{6}) e^{(cos^{2}2( \frac{\pi}{6}))}\)</span></p><p><span class="math-tex">\( \frac{dy}{dx}=-4sin (\frac{\pi}{3})cos( \frac{\pi}{3}) e^{(cos^{2}( \frac{\pi}{3}))}\)</span></p><p><span class="math-tex">\( \frac{dy}{dx}=-4\times{\frac{\sqrt3}{2}}\times \frac{1}{2}\times e^{(\frac{1}{2})^{2}}\)</span></p><p><span class="math-tex">\( \frac{dy}{dx}=-\sqrt3 e^{\frac{1}{4}}\)</span></p><p>Hence <strong><em>a</em></strong> = 3</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>Consider two functions <strong><em>f</em></strong> and <strong><em>g</em></strong> and their derivatives <strong><em>f'</em></strong> and <strong><em>g'</em></strong>. The following table shows the values for the two functions and their derivatives at x = -1, 0, 1</p><table align="center" border="1" cellpadding="1" cellspacing="0" style="width: 400px;"><thead><tr><th scope="row" style="text-align: center;"> </th><th scope="col" style="text-align: center;">-1</th><th scope="col" style="text-align: center;">0</th><th scope="col" style="text-align: center;">1</th></tr></thead><tbody><tr><th scope="row" style="text-align: center;"><strong><em>f(x)</em></strong></th><td style="text-align: center;">-2</td><td style="text-align: center;">1</td><td style="text-align: center;">6</td></tr><tr><th scope="row" style="text-align: center;"><strong><em>g(x)</em></strong></th><td style="text-align: center;">3</td><td style="text-align: center;">-1</td><td style="text-align: center;">1.5</td></tr><tr><th scope="row" style="text-align: center;"><strong><em>f'(x)</em></strong></th><td style="text-align: center;">2</td><td style="text-align: center;">3</td><td style="text-align: center;">4</td></tr><tr><th scope="row" style="text-align: center;"><strong><em>g'(x)</em></strong></th><td style="text-align: center;">4</td><td style="text-align: center;">-2</td><td style="text-align: center;">3</td></tr></tbody></table><p><span class="math-tex">\(h(x) = f\circ g(x)\)</span></p><p>Find <strong><em>h'(0)</em></strong></p></div><div class="q-answer"><p><strong><em>h'(0) </em></strong>= <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></p></div><div class="q-explanation"><p><strong>h'(x) </strong>=<strong> g'(x)f'(g(x))</strong></p><p><strong>h'(0) </strong>=<strong> g'(0)f'(g(0))</strong></p><p><strong>h'(0) </strong>=<strong> g'(0)f'(-1)</strong></p><p><strong>h'(0) </strong>=<strong> -1<span class="math-tex">\(\times\)</span>2</strong></p><p><strong>h'(0) </strong>=<strong> -2</strong></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa 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