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<p><img alt="" src="../../files/differentiation/product-quotient/main-1.jpg" style="float: left; width: 100px; height: 100px;"></p> <p>The Product Rule is a formula that we can use to differentiate the product of 2 (or more) functions. The Quotient Rule is for the quotient of two functions (one function divided by another). The rules are quite easy to apply. The challenge is often simplifying your answer so that you can find the coordinates of any stationary points. To do this you need to do some careful factorising. It is also important to be confident in using other differentiation techniques, like the Chain Rule. </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>the product rule for differentiation</li> <li>the quotient rule for differentiation</li> </ul> <table border="0" cellpadding="0" cellspacing="0" style="width:100%;"> <tbody> <tr> <td>Product Rule</td> <td><span class="math-tex">\(y=uv\\ \frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}\)</span></td> <td><span class="math-tex">\(f(x)=g(x)h(x)\\ f'(x)=g(x)h'(x)+h(x)g'(x)\)</span></td> </tr> <tr> <td> </td> <td> </td> <td> </td> </tr> <tr> <td>Quotient Rule</td> <td><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></td> <td><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></td> </tr> </tbody> </table> <p> </p> </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>Product Rule Easy</p> </div> </div> <div class="panel-body"> <div class="smart-object center" data-id="397"> <p>In the following example we look at using the product rule to find the derivative of a product of 2 simple functions</p> <hr> <p>Given that <span class="math-tex">\(y=x^2lnx\)</span>, find <span class="math-tex">\(\frac{dy}{dx}\)</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/276032834"></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/product-quotient/product_rule_easy.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/product-quotient/product_rule_easy.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel panel-yellow panel-has-border panel-has-colored-body panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Product Rule Hard</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="398"> <p>Applying the product rule is not difficult, but what can be challenging is simplifying the result so that stationary points can be found. In order to do this careful factorising is required. Here is an example that shows how to do it.</p> <hr> <p><span class="math-tex">\(f(x)=x^3(x-3)^2\)</span></p> <p>The function f has three stationary points. Find the x coordinates of these points</p> <hr class="hidden-separator"> <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/276040220"></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/product-quotient/product_rule_hard.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/product-quotient/product_rule_hard.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-border panel-has-colored-body panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Quotient Rule Easy</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="399"> <p>The Quotient Rule is fairly straight forward to apply. However, functions that require this rule often require good knowledge of other differentiation techniques, like the Chain Rule. The example below requires some algebraic simplifying techniques also.</p> <hr> <p>Let <span class="math-tex">\(f(x)=\frac{(3x-2)^2}{x^3} \quad,x\neq 0\)</span></p> <p>Find <span class="math-tex">\(f'(x)\)</span></p> <hr class="hidden-separator"> <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/276059701"></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/product-quotient/quotient_rule_easy.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/product-quotient/quotient_rule_easy.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-expandable panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Quotient Rule Hard</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="400"> <p>One of the challenging things about the Quotient Rule is that we are often required to simplify our answers and leave in them in a factorised form so that we can find the coordinates of any stationary points. Here is a typical example where the simplifying requires careful algebraic manipulation.</p> <hr> <p>The graph of <span class="math-tex">\(y=\frac{2x}{\sqrt{x-1}} \quad ,x>1\)</span> has a local minimum.</p> <p>Find the coordinates of this point.</p> <hr class="hidden-separator"> <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/276066072"></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/product-quotient/quotient_rule_hard.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/product-quotient/quotient_rule_hard.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><iframe align="middle" frameborder="1" height="480" scrolling="yes" src="../../files/differentiation/product-quotient/revision-notes_product_and_quotient_rule.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/differentiation/product-quotient/revision-notes_product_and_quotient_rule.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>Here is a quiz that practises The Product Rule</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#7d30a65d"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="7d30a65d"> <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%;"> Product Rule <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="11-192-743" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Complete the derivatives below with some of the following functions</p><p style="text-align: center;"><span class="q-text-draggable draggable" draggable="true">1</span> <span class="q-text-draggable draggable" draggable="true">2x²</span> <span class="q-text-draggable draggable" draggable="true">lnx</span> <span class="q-text-draggable draggable" draggable="true">x</span> <span class="q-text-draggable draggable" draggable="true">x²</span> <span class="q-text-draggable draggable" draggable="true">2x</span></p></div><div class="q-answer"><p><span class="math-tex">\(\frac{d}{dx}(xe^x)\)</span> = e<sup>x</sup> + e<sup>x</sup><input type="text" style="height: auto;" data-c="x"> <span class="review"></span> </p><p><span class="math-tex">\(\frac{d}{dx}(x^2e^x)\)</span> = x²e<sup>x</sup> + e<sup>x</sup><input type="text" style="height: auto;" data-c="2x"> <span class="review"></span></p><p><span class="math-tex">\(\frac{d}{dx}(xlnx)\)</span> = lnx + <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p><p><span class="math-tex">\(\frac{d}{dx}(x^2lnx)\)</span> = x + 2x <input type="text" style="height: auto;" data-c="lnx"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}\)</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>Complete the derivatives below with some of the following functions</p><p style="text-align: center;"><span class="q-text-draggable draggable" draggable="true">lnx</span> <span class="q-text-draggable draggable" draggable="true">(x - 3)</span> <span class="q-text-draggable draggable" draggable="true">(x - 3)²</span> <span class="q-text-draggable draggable" draggable="true">2(x - 3)</span> <span class="q-text-draggable draggable" draggable="true">x²</span> <span class="q-text-draggable draggable" draggable="true">3x²</span></p></div><div class="q-answer"><p><span class="math-tex">\(\frac{d}{dx}[x^2(x-3)^3]\)</span> = 2x(x - 3)<sup>3</sup> + (x - 3)<sup>2</sup> <input type="text" style="height: auto;" data-c="3x²"> <span class="review"></span></p><p><span class="math-tex">\(\frac{d}{dx}[x^3(x-3)^2]\)</span> = 3x²(x - 3)<sup>2</sup> + 2x<sup>3</sup> <input type="text" style="height: auto;" data-c="(x - 3)"> <span class="review"></span></p><p><span class="math-tex">\(\frac{d}{dx}[e^xlnx]\)</span>= <span class="math-tex">\(\frac{e^x}{x}\)</span> + e<sup>x</sup> <input type="text" style="height: auto;" data-c="lnx"> <span class="review"></span></p><p><span class="math-tex">\(\frac{d}{dx}[\sqrt x (x-3)^2]\)</span> = <span class="math-tex">\(\frac{(x-3)^2}{2\sqrt x}+\sqrt x\)</span> <input type="text" style="height: auto;" data-c="2(x - 3)"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}\)</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">\(\frac{d}{dx}[e^{2x}lnx]=\)</span></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(2e^{2x}lnx+\frac {e^{2x}}{2x}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(e^{2x}lnx+\frac {e^{2x}}{x}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(e^{2x}lnx+\frac {e^{2x}}{2x}\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(2e^{2x}lnx+\frac {e^{2x}}{x}\)</span></span></label> </p></div><div class="q-explanation"><p><span class="math-tex">\(\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}\)</span></p><p><span class="math-tex">\(\frac{d}{dx}(ln2x)=\frac{2}{2x}=\frac{1}{x}\)</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>Consider two functions <strong><em>f</em></strong> and <strong><em>g</em></strong> . Given that <em><strong>h</strong></em>(x) = <em><strong>f</strong></em>(x)<em><strong>g</strong></em>(x)</p><p>The table shows valuse for <strong><em>f </em></strong>,<strong><em> g </em></strong>,<strong><em> h </em></strong>and their derivatives<strong><em> f' </em></strong>,<strong><em> g' </em></strong>and<strong><em> h' </em></strong>for x = 1 and x = 2</p><p>Example,<strong><em> </em></strong><em><strong>f</strong></em>(1) = 2 and <strong><em>g '</em></strong>(2)<strong><em> </em></strong>= -2</p><p>Complete the table</p></div><div class="q-answer"><table align="center" border="1" cellpadding="0" cellspacing="0"><thead><tr><th scope="row" style="text-align: center;">x</th><th scope="col" style="text-align: center;">1</th><th scope="col" style="text-align: center;">2</th></tr></thead><tbody><tr><th scope="row" style="text-align: center;">f(x)</th><td style="text-align: center;">2</td><td style="text-align: center;"> <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></td></tr><tr><th scope="row" style="text-align: center;">f '(x)</th><td style="text-align: center;">3</td><td style="text-align: center;">2</td></tr><tr><th scope="row" style="text-align: center;">g(x)</th><td style="text-align: center;">-1</td><td style="text-align: center;">3</td></tr><tr><th scope="row" style="text-align: center;">g '(x)</th><td style="text-align: center;">1</td><td style="text-align: center;"> <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></td></tr><tr><th scope="row" style="text-align: center;">h(x)</th><td style="text-align: center;"> <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></td><td style="text-align: center;">3</td></tr><tr><th scope="row" style="text-align: center;">h '(x)</th><td style="text-align: center;"> <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span></td><td style="text-align: center;">4</td></tr></tbody></table></div><div class="q-explanation"><p><em><strong>h</strong></em>(1) = <em><strong>f</strong></em>(1)<em><strong>g</strong></em>(1)</p><p><em><strong>h '</strong></em>(1) = <em><strong>f</strong></em> '(1)<em><strong>g</strong></em>(1) <em><strong>+ </strong></em> <em><strong>f</strong></em>(1)<em><strong>g '</strong></em>(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> </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 quiz that practises The Quotient Rule</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#15b9ee55"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="15b9ee55"> <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%;"> Quotient Rule <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="11-193-743" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Complete the following derivative</p><p><span class="math-tex">\(\frac{d}{dx}(\frac {x}{x+3})=\frac { a\cdot 1-x\cdot b }{ { (x+3) }^{ c } } \)</span></p></div><div class="q-answer"><p> <span class="q-text-draggable draggable" draggable="true">x + 3</span> <span class="q-text-draggable draggable" draggable="true">1</span> <span class="q-text-draggable draggable" draggable="true">2</span> <span class="q-text-draggable draggable" draggable="true">x</span> <span class="q-text-draggable draggable" draggable="true">3</span></p><p>a = <input type="text" style="height: auto;" data-c="x + 3"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="1"> <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">\(\frac{d}{dx}(\frac {u}{v})=\frac { v\cdot \frac{du}{dx}-u\cdot \frac{dv}{dx} }{ v^2 } \\ \frac{d}{dx}(\frac {x}{x+3})=\frac { (x+3)\cdot 1-x\cdot 1 }{ { (x+3) }^{ c2} } \)</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>Complete the following derivative</p><p><span class="math-tex">\(\frac{d}{dx}(\frac {e^x}{x})=\frac { a\cdot e^x-e^x }{ b } \)</span></p></div><div class="q-answer"><p> <span class="q-text-draggable draggable" draggable="true">x</span> <span class="q-text-draggable draggable" draggable="true">x²</span> <span class="q-text-draggable draggable" draggable="true">1</span> <span class="q-text-draggable draggable" draggable="true">e<sup>x</sup></span></p><p>a = <input type="text" style="height: auto;" data-c="x"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="x²"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\frac{d}{dx}(\frac {u}{v})=\frac { v\cdot \frac{du}{dx}-u\cdot \frac{dv}{dx} }{ v^2 } \\
\frac{d}{dx}(\frac {e^x}{x})=\frac { x\cdot e^x-e^x }{ x² } \)</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>Complete the following derivative</p><p><span class="math-tex">\(\frac{d}{dx}(\frac {lnx}{x})=\frac { a - 1 \cdot b }{ c } \)</span></p></div><div class="q-answer"><p> <span class="q-text-draggable draggable" draggable="true">1</span> <span class="q-text-draggable draggable" draggable="true">lnx</span> <span class="q-text-draggable draggable" draggable="true">x²</span> <span class="q-text-draggable draggable" draggable="true">x</span> <span class="q-text-draggable draggable" draggable="true">xlnx</span></p><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="lnx"> <span class="review"></span></p><p>c = <input type="text" style="height: auto;" data-c="x²"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\frac{d}{dx}(\frac {u}{v})=\frac { v\cdot \frac{du}{dx}-u\cdot \frac{dv}{dx} }{ v^2 } \\
\frac{d}{dx}(\frac {lnx}{x})=\frac { x \cdot \frac{1}{x} - 1 \cdot lnx }{ x^2 } \)</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>Given that <span class="math-tex">\(f(x) =\frac{x^2-1}{x^2+1}\)</span> , then <span class="math-tex">\(f'(x) =\frac{a}{(x^2+1)^2}\)</span></p><p>Find <em><strong>a </strong></em>in its simplest form</p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="4x"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(f'(x) =\frac{2x(x^2+1)-2x(x^2-1)}{(x^2+1)^2}\\
f'(x) =\frac{2x^3+2x-2x^3+2x}{(x^2+1)^2}\\
f'(x) =\frac{4x}{(x^2+1)^2}\)</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 the value of <strong><em>a </em></strong> in the following derivative</p><p><span class="math-tex">\(\frac{d}{dx}(\frac {x}{x-1})=\frac { a }{ (x-1)^2} \)</span></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\frac{d}{dx}(\frac {x}{x-1})=\frac { (x-1) \cdot 1-x \cdot1 }{ (x-1)^2} \\
\quad \quad \quad \quad=\frac { x-1-x }{ (x-1)^2}\\
\quad \quad \quad \quad=\frac { -1 }{ (x-1)^2}\)</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 the value of <strong><em>a </em></strong> in the following derivative</p><p><span class="math-tex">\(\frac{d}{dx}(\frac {2e^x-1}{2e^x+1})=\frac { ae^x }{ (2e^x+1)^2} \)</span></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\frac{d}{dx}(\frac {2e^x-1}{2e^x+1})=\frac { (2e^x+1)2e^x-2e^x(2e^x-1) }{ (2e^x+1)^2} \\
\quad \quad \quad \quad=\frac { 4e^{2x}+2e^x-4e^{2x}+2e^x }{ (2e^x+1)^2} \\
\quad \quad \quad \quad=\frac { 4e^x }{ (2e^x+1)^2} \\\)</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>Consider two functions <strong><em>f</em></strong> and <strong><em>g</em></strong> . Given that <span class="math-tex">\(h(x)=\frac{f(x)}{g(x)}\)</span></p><p>The table shows valuse for <strong><em>f </em></strong>,<strong><em> g </em></strong>,<strong><em> h </em></strong>and their derivatives<strong><em> f' </em></strong>,<strong><em> g' </em></strong>and<strong><em> h' </em></strong>for x = 1 and x = 2</p><p>Example,<strong><em> </em></strong><em><strong>f</strong></em>(1) = 3 and <strong><em>g '</em></strong>(2)<strong><em> </em></strong>= -2</p><p>Complete the table</p></div><div class="q-answer"><table align="center" border="1" cellpadding="0" cellspacing="0"><thead><tr><th scope="row" style="text-align: center;">x</th><th scope="col" style="text-align: center;">1</th><th scope="col" style="text-align: center;">2</th></tr><tr><th scope="row" style="text-align: center;">f(x)</th><td style="text-align: center;">3</td><td style="text-align: center;"> <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></td></tr><tr><th scope="row" style="text-align: center;">f '(x)</th><td style="text-align: center;">-2</td><td style="text-align: center;">2</td></tr><tr><th scope="row" style="text-align: center;">g(x)</th><td style="text-align: center;">-1</td><td style="text-align: center;">1</td></tr><tr><th scope="row" style="text-align: center;">g '(x)</th><td style="text-align: center;">1</td><td style="text-align: center;"> <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></td></tr><tr><th scope="row" style="text-align: center;">h(x)</th><td style="text-align: center;"> <input type="text" style="height: auto;" data-c="-3"> <span class="review"></span></td><td style="text-align: center;">2</td></tr><tr><th scope="row" style="text-align: center;">h '(x)</th><td style="text-align: center;"> <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span></td><td style="text-align: center;">6</td></tr></thead></table></div><div class="q-explanation"><p><span class="math-tex">\(h(1)=\frac{f(1)}{g(1)}=\frac{3}{-1}=-3\)</span></p><p><span class="math-tex">\(h'(1)=\frac{g(1)f'(1)-g'(1)f(1)}{[g(1)]^2}=\frac{-1(-2)-1\cdot3}{[-1]^2}=-1\)</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> </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="395"> <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>Let <span class="math-tex">\(y=xe^x\)</span></p> <p>a) Find <span class="math-tex">\(\frac{dy}{dx}\)</span></p> <p>b) Show that <span class="math-tex">\(\frac{d^2y}{dx^2}=e^x(2+x)\)</span></p> <p>c) Find the coordinates of the stationary point and show that it is a local minimum.</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>a) Use the Product Rule to find <span class="math-tex">\(\frac{dy}{dx}\)</span><content></content></p> <p>c) Solve <span class="math-tex">\(\frac{dy}{dx}=0\)</span> to find position of stationary point. Show that <span class="math-tex">\(\frac{d^2y}{dx^2}>0\)</span></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/product-quotient/esq_differentiation_product-quotient_rule1a.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/product-quotient/esq_differentiation_product-quotient_rule1a.pdf" width="640"></iframe></p> </section> <h4> </h4> </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="392"> <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>Let <span class="math-tex">\(f(x)={ x }^{ 2 }{ (2x-3) }^{ 3 }\)</span></p> <p>a) Find <span class="math-tex">\(f'(x)\)</span></p> <p>b) The graph of y = f(x) has stationary points at x = 0, x= <span class="math-tex">\(\frac{3}{2}\)</span> and x = <strong>a</strong>. Find the value of <strong>a</strong></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>a) You need to use the chain rule to differentiate <span class="math-tex">\({ (2x-3) }^{ 3 }\)</span></p> <p><content>b) Factorise your answer to part a)</content></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/product-quotient/esq_differentiation_product-quotient_rule1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/product-quotient/esq_differentiation_product-quotient_rule1.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-border panel-has-colored-body panel-expandable panel-default"> <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="394"> <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>Let <span class="math-tex">\(f(x)=\frac{lnx}{x},x>0\)</span></p> <p>a) Show that <span class="math-tex">\(f'(x)=\frac{1-lnx}{x^2}\)</span></p> <p>b) Find <span class="math-tex">\(f''(x)\)</span></p> <p>c) The graph of <em>f</em> has a point of inflexion at A. Find the x-coordinate of A.</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>a) Use the Quotient Rule to find <span class="math-tex">\(f'(x)\)</span><content></content></p> <p>c) Solve <span class="math-tex">\(f''(x)=0\)</span> to find point of inflexion</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/product-quotient/esq_differentiation_product-quotient_rule3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/product-quotient/esq_differentiation_product-quotient_rule3.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-border panel-has-colored-body panel-expandable panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="393"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Let <span class="math-tex">\(f(x)=tanx\)</span>. A<span class="math-tex">\((\frac{\pi}{3},\sqrt{3})\)</span> is a point that lies on the graph of <span class="math-tex">\(y=f(x)\)</span></p> <p>a) Given that <span class="math-tex">\(tanx=\frac{sinx}{cosx}\)</span> find <span class="math-tex">\(f'(x)\)</span></p> <p>b) Show that <span class="math-tex">\(f'(\frac{\pi}{3})=4\)</span></p> <p>c) Find the equation of the normal to the curve y = f(x) at the point A</p> <p>d) Show that the normal crosses the y axis at <span class="math-tex">\(\sqrt{3}+\frac{\pi}{12}\)</span></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"> <p>a) Use the quotient rule to differentiate tanx</p> <p><content>c) <span class="math-tex">\(gradient \ of \ normal =-\frac{1}{gradient \ of \ tangent}\)</span></content></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/product-quotient/esq_differentiation_product-quotient_rule2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/product-quotient/esq_differentiation_product-quotient_rule2.pdf" width="640"></iframe></p> </section> <h4> </h4> </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 5</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="396"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Let <span class="math-tex">\(f(x)=e^{2x}cosx\)</span></p> <p>a) Find <span class="math-tex">\(f'(x)\)</span></p> <p>b) Show that <span class="math-tex">\(f''(x)=4f'(x)-5f(x)\)</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"> <p>a) Use the Product Rule to find <span class="math-tex">\(f'(x)\)</span></p> <p>b) Use Product Rule again to find <span class="math-tex">\(f''(x)\)</span></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/product-quotient/esq_differentiation_product-quotient_rulehl5.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/differentiation/product-quotient/esq_differentiation_product-quotient_rulehl5.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="743"> <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>Product and Quotient Rule</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 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