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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"> Introducing Derivatives <a href="#" class="mark-page-favorite pull-right" data-pid="657" 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">Introducing Derivatives</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/introduction/main-1.jpg" style="float: left; width: 100px; height: 100px;"></p> <p><img class="sibico" src="../../../img/sibico/hl-blue.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL"> <img class="sibico" src="../../../img/sibico/sl-blue.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL"> Differentiation allows us to find <strong>rates of change</strong>. The <strong>derivative </strong>is the rate at which one quantity changes with respect to another. When we <strong>differentiate</strong> a function we find the <strong>gradient function</strong>. On a graph, the <strong>derivative</strong> is the <strong>slope</strong> of the curve at any point on the curve.</p> <p>There are many practical applications of differentiation that you will meet in your IB subjects and later in your professional career</p> <hr class="hidden-separator"> <ul> <li>acceleration = rate of change of velocity</li> <li>population growth = rate of population</li> <li>force applied to a body = rate of change of momentum</li> <li>marginal cost = rate of change of cost</li> <li>chemical reaction rate = rate of change of concentration</li> </ul> <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>rates of change</li> <li>gradient function</li> <li>differentiation from first principles</li> <li>different notation for derivatives</li> <li>the power rule to differentiate <span class="math-tex">\(ax^n\)</span></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>Definition of a derivative</p> </div> </div> <div class="panel-body"> <div class="smart-object center" data-id="373"> <p>The graph of the function f(x) = x² is shown below.</p> <p>This applet calculates an <strong>approximation</strong> of the gradient of f(x) at A. You should notice that the secant AB is steeper than the gradient of the graph at A. However, you can improve the approximation by dragging the point B closer to A. The closer the point B moves to A, the better and better the approximation becomes. Try it for yourself!</p> <p>This is the basis of where the definition of the gradient function comes from.</p> <p style="text-align: center"><iframe height="500px" scrolling="no" src="https://www.geogebra.org/material/iframe/id/AVs94wwK/width/650/height/500/border/888888/smb/false/stb/true/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/true/sdz/true/ctl/false" style="border:0px;" title="differentiation first principles" width="650px"></iframe></p> <p>The graph below shows the general function f(x) = <span class="math-tex">\(x^n\)</span></p> <p>As before, the secant gives an approximation of the gradient of the curve.</p> <p>As the two points get closer and closer, the approximation gets better and better.</p> <p><img alt="" src="../../files/differentiation/introduction/first-principles.jpg" style="width: 300px; height: 325px;"></p> <p>If we could make this secant infinitesimally small, then the gradient of the secant would equal the gradient of the curve at this point.</p> <p>The <strong>limit </strong>of the gradient of an infinitesimally small secant gives the gradient of the curve. We can describe this using the following formula</p> <p style="text-align: center;"><span class="math-tex">\(f'(x)=\lim\limits_{h \to 0} \frac{f(x+h)-f(x)}{h}\)</span></p> <p>The process of finding the gradient function like this is sometimes called <strong>differentiation from first principles. </strong>In the following video, we will look at using this formula to find the gradient function for f(x) = x²</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/272369852"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/differentiation/introduction/differentiation_from_1st_principles.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/introduction/differentiation_from_1st_principles.pdf" width="640"></iframe></p> </section> </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>Notation</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="903"> <p>There are several ways of writing derivatives:</p> <table border="1" cellpadding="0" cellspacing="0" height="74" width="490"> <tbody> <tr> <td><span class="math-tex">\(y = x²\)</span></td> </tr> <tr> <td><span class="math-tex">\(\frac { dy }{ dx } =2x\)</span></td> </tr> <tr> <td> </td> </tr> <tr> <td> </td> </tr> <tr> <td><span class="math-tex">\(f(x) = x²\)</span></td> </tr> <tr> <td><span class="math-tex">\(f'(x)=2x\)</span></td> </tr> <tr> <td> </td> </tr> <tr> <td> </td> </tr> <tr> <td><span class="math-tex">\(\frac { d }{ dx }(x^2) =2x\)</span></td> </tr> </tbody> </table> <h3>Power Rule</h3> <p>We can find a rule for differentiating functions in the form y = x<sup>n</sup></p> <p><img alt="" class="gifffer" data-gifffer="/media/mathsanalysis/files/differentiation/introduction/power_rule.gif"></p> <table align="center" border="2" cellpadding="5" cellspacing="5"> <thead> <tr> <th scope="col">function</th> <th scope="col">gradient</th> </tr> </thead> <tbody> <tr> <td>y = x</td> <td><span class="math-tex">\(\frac { dy }{ dx } =1\)</span></td> </tr> <tr> <td> </td> <td> </td> </tr> <tr> <td>y = x<sup>2</sup></td> <td><span class="math-tex">\(\frac { dy }{ dx } =2x\)</span></td> </tr> <tr> <td> </td> <td> </td> </tr> <tr> <td>y =<sup> </sup>x<sup>3</sup></td> <td><span class="math-tex">\(\frac { dy }{ dx } =3x^2\)</span></td> </tr> <tr> <td> </td> <td> </td> </tr> <tr> <td>y =x<sup>4</sup></td> <td><span class="math-tex">\(\frac { dy }{ dx } =4x^3\)</span></td> </tr> <tr> <td> </td> <td> </td> </tr> <tr> <td>y =x<sup>n</sup></td> <td><span class="math-tex">\(\frac { dy }{ dx } =nx^{n-1}\)</span></td> </tr> </tbody> </table> <p>Also</p> <p>1) if <span class="math-tex">\(y=a\cdot x^n\)</span> then <span class="math-tex">\(\frac { dy }{ dx } =a\cdot nx^{n-1}\)</span></p> <p>2) if y = f(x) + g(x) then <span class="math-tex">\(\frac { dy }{ dx } =f'(x) + g'(x)\)</span></p> </div> </div> </div> <div class="panel-footer"> <div> </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>Power Rule</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="375"> <p>The video below goes through some examples using the power rule <span></span><span class="math-tex">\(\frac { d }{ dx }(ax^n) =anx^{n-1}\)</span><span> and all the different types of notation:</span></p> <p><em>Differentiate the following with respect to x</em></p> <p><em>1) <span class="math-tex">\(y=6{ x }^{ 3 }\)</span></em></p> <p>2) <span class="math-tex">\(f(x)=\frac { 3 }{ { x }^{ 2 } } \)</span></p> <p>3) <span class="math-tex">\(\frac { 5 }{ \sqrt { { x }^{ 3 } } } \)</span></p> <p>4) <span class="math-tex">\(y=\frac { 5x-2 }{ \sqrt { x } } \)</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/272439055"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/differentiation/introduction/the-power-rule.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/introduction/the-power-rule.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </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>Gradient at a Point</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="374"> <p>In the video below, we look at the following example</p> <p><em>Find the gradient of the curve <span class="math-tex">\(y=\frac { 3 }{ { x }^{ 2 } } +2\sqrt { x } -3\)</span> at the point (1,2)</em></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/272432998"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/differentiation/introduction/find-gradient-of-curve-at-a-point.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/introduction/find-gradient-of-curve-at-a-point.pdf" width="640"></iframe></p> </section> </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 style="text-align: center;"><iframe align="middle" frameborder="1" height="480" scrolling="yes" src="../../files/differentiation/introduction/revision-notes_the-derivative.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/differentiation/introduction/revision-notes_the-derivative.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>The Power Rule for differentiating functions in the form f(x) = ax<sup>n </sup>is not difficult to use. Often the tricky thing is to understand the indices in the function. This quiz gives you some practice in manipulating indices. Fill in the gaps</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#dab0523f"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="dab0523f"> <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%;"> Indices Equivalence 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="11-187-657" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p><span class="math-tex">\(\frac { 1 }{ { x }^{ 2 } } ={ x }^{ a }\)</span>.</p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\({ x }^{ n }=\frac { 1 }{ { x }^{ -n } } \)</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">\(\sqrt { x } ={ x }^{ a }\)</span></p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="0.5"> <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">\(\sqrt [ 3 ]{ x } ={ x }^{ \frac { a }{ b } }\)</span></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\sqrt [ n ]{ x } ={ x }^{ \frac { 1 }{ n } }\)</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">\(\sqrt { { x }^{ 3 } } ={ x }^{ a }\)</span></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="1.5"> <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">\(\sqrt [ 3 ]{ { x }^{ 2 } } =\quad { x }^{ \frac { a }{ b } }\)</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="3"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\sqrt [ n ]{ { x }^{ m } } =\quad { x }^{ \frac { m }{ n } }\)</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">\(\frac { 1 }{ \sqrt { x } } ={ x }^{ a }\)</span></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="-0.5"> <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">\(\frac { 4 }{ \sqrt { { x }^{ 3 } } } =a{ x }^{ \frac { b }{ 2 } }\)</span></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span> </p><p>b = <input type="text" style="height: auto;" data-c="-3"> <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">\(x\sqrt { x } ={ x }^{ \frac { a }{ b } }\)</span></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="3"> <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"><p><span class="math-tex">\(x\sqrt { x } = x\cdot{ x }^{ \frac { 1 }{ 2 } }={ x }^{ 1.5 }\)</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><span class="math-tex">\(\frac { 3{ x }^{ 2 } }{ \sqrt { x } } =a{ x }^{ \frac { b }{ c } }\)</span></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p><p>c = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\frac { { 3x }^{ 2 } }{ \sqrt { x } } =\frac { { 3x }^{ 2 } }{ { x }^{ 0.5 } } ={ 3x }^{ 2-0.5 }={ 3x }^{ 1.5 }\)</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><span class="math-tex">\(\frac { 2x({ x }^{ 2 }+3) }{ \sqrt { x } } =2{ x }^{ \frac { a }{ b } }+6{ x }^{ \frac { 1 }{ 2 } }\)</span></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="5"> <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> </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>The following quiz gives you some practice in using the Power Rule</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#4fa46290"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="4fa46290"> <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 <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="11-188-657" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Find the gradients of the following functions</p></div><div class="q-answer"><p>f(x) = 2x + 3 , gradient = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><p>y = -3x + 6 , gradient = <input type="text" style="height: auto;" data-c="-3"> <span class="review"></span></p><p>y = 7 , gradient = <input type="text" style="height: auto;" data-c="0"> <span class="review"></span></p></div><div class="q-explanation"><p>The graphs of all of the functions are straight line in the form y = ax + b</p><p>y = 7 is a horizontal line, so the gradient = 0</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 gradient function of y = 3x<sup>4</sup> is <span class="math-tex">\( \frac { dy }{ dx } =a{ x }^{ b }\)</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="3"> <span class="review"></span></p></div><div class="q-explanation"><p>The gradient function of y = ax<sup>n</sup> is <span class="math-tex">\( \frac { dy }{ dx } =an{ x }^{ n-1 }\)</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>The gradient function of f(x) = <span class="math-tex">\(\\ \sqrt { x } \)</span> is <span class="math-tex">\(f'(x)=a{ x }^{ b }\)</span></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="0.5"> <span class="review"></span></p><p>b = <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">\(\sqrt { x } ={ x }^{ 0.5 }\)</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><span class="math-tex">\(\frac { d }{ dx } \left( \sqrt [ 3 ]{ { x }^{ 2 } } \right) =\frac { 2 }{ a } { x }^{ \frac { -1 }{ b } }\)</span></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> </p></div><div class="q-explanation"><p><span class="math-tex">\(\frac { d }{ dx } \left( \sqrt [ 3 ]{ { x }^{ 2 } } \right) =\frac { d }{ dx } \left( { x }^{ \frac { 2 }{ 3 } } \right) =\frac { 2 }{ 3 } { x }^{ \frac { 2 }{ 3 } -1 }=\frac { 2 }{ 3 } { x }^{ \frac { -1 }{ 3 } }\)</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>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="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>Which of the following is the gradient function of <span class="math-tex">\(f(x)=2\sqrt { x } +\frac { 1 }{ x } \)</span></p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(f'(x)=\frac { 1 }{ \sqrt { x } } -\frac { 1 }{ { x }^{ 2 } } \)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(f'(x)={ \sqrt { x } } +\frac { 1 }{ { x }^{ 2 } } \)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(f'(x)=\frac { 1 }{ \sqrt { x } } -x^0\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(f'(x)={ \sqrt { x } } +x^0\)</span></span></label> </p></div><div class="q-explanation"><p><span class="math-tex">\(f(x)=2\sqrt { x } +\frac { 1 }{ x } =2{ x }^{ 0.5 }+{ x }^{ -1 }\\ \Rightarrow f'(x)=2\cdot (0.5){ x }^{ -0.5 }\quad +(-1){ x }^{ -2 }\quad =\quad \frac { 1 }{ { x }^{ 0.5 } } -\frac { 1 }{ { x }^{ 2 } } \)</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><span class="math-tex">\(\frac { d }{ dx } \left( { 2x }^{ 5 }-3{ x }^{ 2 }-5 \right) =a{ x }^{ b }-c{ x }^{ d }+e\)</span></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="10"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p><p>c = <input type="text" style="height: auto;" data-c="6"> <span class="review"></span></p><p>d = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p><p>e = <input type="text" style="height: auto;" data-c="0"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\frac { d }{ dx } \left( { 2x }^{ 5 }-3{ x }^{ 2 }-5 \right) =10{ x }^{ 4 }-6{ x }^{ 1 }+0\)</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 class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(y=\frac { 2{ x }^{ 3 }+4x }{ x } \)</span> then <span class="math-tex">\(\frac { dy }{ dx } =a{ x }^{ b }\)</span></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(y=\frac { 2{ x }^{ 3 }+4x }{ x } =\frac { x(2{ x }^{ 2 }+4) }{ x } =2{ x }^{ 2 }+4\\ \Rightarrow \frac { dy }{ dx } =4x+0\)</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>If <span class="math-tex">\(y=3{ x }^{ 2 }+2\sqrt { x } \)</span> then find <span class="math-tex">\(\frac { dy }{ dx } \)</span> when x = 4</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="24.5"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(y=3{ x }^{ 2 }+2\sqrt { x } =3{ x }^{ 2 }+2{ x }^{ \frac { 1 }{ 2 } }\\ \Rightarrow \frac { dy }{ dx } =\quad 6x+{ x }^{ -\frac { 1 }{ 2 } }=6x+\frac { 1 }{ \sqrt { x } } \\ when\quad x=4\quad ,\quad \frac { dy }{ dx } =24+\frac { 1 }{ \sqrt { 4 } } =24+\frac { 1 }{ 2 } \)</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>Find <span class="math-tex">\(f'(1)\)</span> if <span class="math-tex">\(f(x)=\frac { 3x-1 }{ \sqrt { x } } \)</span></p></div><div class="q-answer"><p>f '(1) <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(f(x)=\frac { 3x-1 }{ \sqrt { x } } =3\sqrt { x } -\frac { 1 }{ \sqrt { x } } =3{ x }^{ 0.5 }-{ x }^{ -0.5 }\\ f'(x)=3\cdot 0.5{ x }^{ -0.5 }-(-0.5){ x }^{ -1.5 }\\ f'(x)=\frac { 3 }{ 2\sqrt { x } } +\frac { 1 }{ 2\sqrt { { x }^{ 3 } } } \\ f'(1)=\frac { 3 }{ 2\sqrt { 1 } } +\frac { 1 }{ 2\sqrt { { 1 }^{ 3 } } } \\ f'(1)=\frac { 3 }{ 2 } +\frac { 1 }{ 2 } =2\)</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>y = x<sup>2</sup>-6x</p><p>Find the value of x where the gradient function = 0</p></div><div class="q-answer"><p>x = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\frac { dy }{ dx } =2x-6\\ 2x-6=0\\ x=3\)</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> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Exam-style Questions</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="377"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL easy"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Find the value(s) of x for which the graph <span class="math-tex">\(y=x^3-8x+2\)</span> has gradient 4</p> <h4>Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">Find where <span class="math-tex">\(\frac{dy}{dx}=4\)</span><content></content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/differentiation/introduction/esq_differentiation_power_rule1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/introduction/esq_differentiation_power_rule1.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="376"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Find <span class="math-tex">\(f'(4)\)</span> for the function <span class="math-tex">\(f(x)=2x+\frac { 8 }{ \sqrt { x } } +\frac { 32 }{ x } \)</span></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"><span class="math-tex">\(\frac { 1 }{ \sqrt { x } } ={ x }^{ -\frac { 1 }{ 2 } }\)</span><content> </content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/differentiation/introduction/esq_differentiation_power_rule2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/introduction/esq_differentiation_power_rule2.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="378"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The gradient of <em><span class="math-tex">\(y=x^2+ax+b\)</span></em> at the point (1,-3) is -4.</p> <p>Find <em><strong>a</strong></em> and <em><strong>b</strong></em></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content>We know 2 facts about this graph:</p> <p>x=1 when y = -3</p> <p><span class="math-tex">\(\frac{dy}{dx}=-4\)</span> when x = 1</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/differentiation/introduction/esq_differentiation_power_rule3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/introduction/esq_differentiation_power_rule3.pdf" width="640"></iframe></p> </section> </div> </div> </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="657"> <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>Introducing Derivatives</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 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