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<!DOCTYPE html>
<html data-require="math calculus polynomials expressions">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Product rule</title>
<script src="../khan-exercise.js"></script>
</head>
<body>
<div class="exercise">
<div class="vars" data-ensure="FUNCN.fText !== FUNCD.fText">
<var id="FUNCN">generateSpecialFunction("x")</var>
<var id="FUNCD">generateSpecialFunction("x")</var>
<var id="NOTATION">funcNotation("x")</var>
</div>
<div class="problems">
<div>
<div class="question">
<p>Find <code>\displaystyle \frac{d}{dx} \bigl((<var>FUNCN.fText</var>)(<var>FUNCD.fText</var>)\bigr)</code>.</p>
</div>
<p class="solution"><code>\lrsplit{(<var>FUNCN.ddxFText</var>)(<var>FUNCD.fText</var>)}{+\:(<var>FUNCD.ddxFText</var>)(<var>FUNCN.fText</var>)}</code></p>
<ul class="choices" data-show="5" data-none="true">
<!-- 5 = 3 incorrect + 1 correct + "none" -->
<li><code>\lrsplit{(<var>FUNCD.fText</var>)(<var>FUNCN.ddxFText</var>)}{-\:(<var>FUNCN.fText</var>)(<var>FUNCD.ddxFText</var>)}</code></li>
<li><code>\lrsplit{(<var>FUNCN.fText</var>)(<var>FUNCN.ddxFText</var>)}{+\:(<var>FUNCD.fText</var>)(<var>FUNCD.ddxFText</var>)}</code></li>
<li><code>\lrsplit{(<var>FUNCN.fText</var>)(<var>FUNCD.ddxFText</var>)}{-\:(<var>FUNCD.fText</var>)(<var>FUNCN.ddxFText</var>)}</code></li>
</ul>
</div>
</div>
<div class="hints">
<p>We know <code>\displaystyle \frac{d}{dx} \bigl( f(x) g(x) \bigr) = f'(x)g(x) + g'(x)f(x)</code>.</p>
<div>
<p>In this case,</p>
<p><code>\qquad f(x) = <var>FUNCN.fText</var></code>,</p>
<p><code>\qquad g(x) = <var>FUNCD.fText</var></code>.</p>
</div>
<p>Differentiate each function:</p>
<p><code>\qquad f'(x) = <var>FUNCN.ddxFText</var></code>,</p>
<p><code>\qquad g'(x) = <var>FUNCD.ddxFText</var></code>.</p>
<div>
<p>Thus, the answer is</p>
<p><code>\qquad {(<var>FUNCN.ddxFText</var>)(<var>FUNCD.fText</var>) + (<var>FUNCD.ddxFText</var>)(<var>FUNCN.fText</var>)}</code>.</p>
</div>
</div>
</div>
</body>
</html>
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