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    <name>
      <text>11.7.2a first imp derivative</text>
    </name>
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      <text><![CDATA[<p class="HELM_exercise">Exercise</p>
<p>Use implicit differentiation to find&nbsp; \[\frac{dy}{dx}\] <br>for the graph of&nbsp; \[x^{@n1@}+y^{@n2@}={@n3@}xy.\]</p>
<p>\( \dfrac{dy}{dx} =\) [[input:ans1]] [[validation:ans1]]</p>]]></text>
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      <text><![CDATA[<p>To find \(\dfrac{dy}{dx}\) for the graph of \(x^{@n1@}+y^{@n2@}={@n3@}xy\) rearrange the equation to give:</p>
<p>\[x^{@n1@}+y^{@n2@}-{@n3@}xy =0.\]</p>
<p>Thus:</p>
<p>\[\frac{d}{dx}\left(x^{@n1@}+y^{@n2@}-{@n3@}xy\right) =0\]</p>
<p>\[{@n1@}{@x^(n1-1)@}+{@n2@}{@y^(n2-1)@}\frac{dy}{dx}-{@n3@}(y+x\frac{dy}{dx})=0\]</p>
<p>\[\frac{dy}{dx}({@n2@}{@y^(n2-1)@}-{@n3@}x) = {@n3@}y-{@n1@}{@x^(n1-1)@}\]</p>
<p>\[\frac{dy}{dx} = \frac{{@n3@}y-{@n1@}{@x^(n1-1)@}}{{@n2@}{@y^(n2-1)@}-{@n3@}x}.\]</p>]]></text>
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      <text>n1:rand(4)+2;
 n2:rand(4)+2;
 n3:rand(8)+2;
 p : x^n1+y^n2=n3*x*y; 
 diffp : expand(n3*y-n1*x^(n1-1))/(n2*y^(n2-1)-n3*x);
 correctanswer:  diffp;
 wronganswer1:n1*x^(n1-1)-n3*y;</text>
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      <text>\(x^{@n1@}+y^{@n2@}={@n3@}xy, \, \frac{dy}{dx} = {@diffp@}\)</text>
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      <text><![CDATA[<p><span class="incorrect">Incorrect answer.</span></p>]]></text>
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