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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>Quotient 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" data-ensure="FUNCN.hints == null || FUNCN.hints.length <= 4">generateSpecialFunction("x")</var>
<var id="FUNCD" data-ensure="FUNCD.hints == null || FUNCD.hints.length <= 4">generateSpecialFunction("x")</var>
<var id="N_F">FUNCN.fText</var>
<var id="N_DF">FUNCN.ddxFText</var>
<var id="D_F">FUNCD.fText</var>
<var id="D_DF">FUNCD.ddxFText</var>
<var id="NOTATION">funcNotation("x")</var>
<var id="ANSWER">function( a, b, c, d, e, min ) {
// create fraction of the form ( (a)(b) + (c)(d) ) / e^2
var term1 = "(" + a + ")" + ( a === b ? "^2" : "(" + b + ")" ),
term2 = "(" + c + ")" + ( c === d ? "^2" : "(" + d + ")" );
return "\\dfrac{" + term1 + min + term2 + "}{(" + e + ")^2}";
}</var>
</div>
<div class="problems">
<div>
<p class="question">Find <code>\displaystyle \frac{d}{dx}\biggl( \frac{<var>FUNCN.fText</var>}{<var>FUNCD.fText</var>} \biggr)</code>.</p>
<!-- (n-df)(d-f) - (d-df)(n-f) / d-f -->
<p class="solution"><code><var>ANSWER( N_DF, D_F, D_DF, N_F, D_F, "-" )</var></code></p>
<ul class="choices" data-show="5" data-none="true">
<!-- (n-df)(d-df) - (d-f)(n-f) / d-f -->
<li data-if="D_DF !== D_F"><code><var>ANSWER( N_DF, D_DF, D_F, N_F, D_F, "-" )</var></code></li>
<!-- (n-df)(d-f) - (d-df)(n-f) / n-f -->
<li data-if="N_F !== D_F"><code><var>ANSWER( N_DF, D_F, D_DF, N_F, N_F, "-" )</var></code></li>
<!-- (n-df)(d-df) - (d-f)(n-f) / n-f -->
<li data-if="N_F !== D_F && D_DF !== D_F"><code><var>ANSWER( N_DF, D_DF, D_F, N_F, N_F, "-" )</var></code></li>
<!-- (n-df)(d-f) + (d-df)(n-f) / d-f -->
<li><code><var>ANSWER( N_DF, D_F, D_DF, N_F, D_F, "+" )</var></code></li>
<!-- (n-df)(d-df) + (d-f)(n-f) / d-f -->
<li data-if="D_DF !== D_F"><code><var>ANSWER( N_DF, D_DF, D_F, N_F, D_F, "+" )</var></code></li>
<!-- (n-df)(d-f) + (d-df)(n-f) / n-f -->
<li data-if="N_F !== D_F"><code><var>ANSWER( N_DF, D_F, D_DF, N_F, N_F, "+" )</var></code></li>
<!-- (n-df)(d-df) + (d-f)(n-f) / n-f -->
<li data-if="N_F !== D_F && D_DF !== D_F"><code><var>ANSWER( N_DF, D_DF, D_F, N_F, N_F, "+" )</var></code></li>
</ul>
</div>
</div>
<div class="hints">
<p>Using the chain rule and the product rule, we know <code>\displaystyle \frac{d}{dx\strut}\frac{f(x)}{g(x)} = \frac{f'(x)g(x) - g'(x)f(x)}{g(x){}^2}</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 \dfrac{{(<var>FUNCN.ddxFText</var>)(<var>FUNCD.fText</var>) - (<var>FUNCD.ddxFText</var>)(<var>FUNCN.fText</var>)}}{(<var>FUNCD.fText</var>)^2}</code>.</p>
</div>
</div>
</div>
</body>
</html>
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