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<quiz>
  <question type="stack">
    <name>
      <text>Diff_Chain_2</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p>Find \[\frac{dy}{dx}\] when \[y=\cosh^{-1}\left(\sqrt{{@n1@} {@v@} + {@n2@}}\right) .\]</p>
<p> [[input:ans1]] [[validation:ans1]]</p>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p>To find \(\frac{dy}{dx}\) when \[y=\cosh^{-1}\left(\sqrt{{@n1@} {@v@} + {@n2@}}\right) .\]</p>
<p>In order to solve this problem we will have to use the chain rule \(\frac{df}{d{@v@}}=\frac{df}{dg} \cdot\frac{dg}{dh}\cdot\frac{dh}{d{@v@}}\)  as follows:</p>
<p>Define \(h({@v@})={@n1@}{@v@}+{@n2@}\), \(g(h) = \sqrt{h} = h^{\frac{1}{2}}\) and \(f(g)=\cosh^{-1}\left(g\right)\).</p>
<p>So, \[\begin{array}{rcl}\displaystyle \frac{d}{d{@v@}}\left(\cosh^{-1}\left(\sqrt{{@n1@} {@v@} + {@n2@}}\right)\right) &amp;\displaystyle =&amp;\displaystyle \frac{df}{dg}\cdot \frac{dg}{dh}\cdot \frac{dh}{d{@v@}} \\ &amp;\displaystyle =&amp;\displaystyle \frac{d}{dg}(\cosh^{-1}\left(g\right))\cdot\frac{d}{dh}(h^{\frac{1}{2}})\cdot \frac{d}{d{@v@}}({@n1@} {@v@}+{@n2@}) \\ &amp;\displaystyle =&amp;\displaystyle \frac{1}{ \sqrt{g^2-1}}\cdot\frac{1}{2}(h^{-\frac{1}{2}})\cdot {@n1@} \\ &amp;\displaystyle =&amp;\displaystyle \frac{1}{ \sqrt{h-1}}\cdot\frac{@n1@}{2 \sqrt{h}} \\&amp;\displaystyle =&amp;\displaystyle {@correctanswer@}\end{array}\]</p>]]></text>
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    <questionvariables>
      <text>v:x;
 n1:rand(8)+2;
 n2:rand(5)+2;
 
 f : acosh(sqrt(n1*v+n2)); 
 g: sqrt(n1*v+n2);
 correctanswer : (n1/2)/(sqrt(n1*v+n2)*sqrt(n1*v+n2-1));
 wronganswer1:(1/2)/(sqrt(n1*v+n2)*sqrt(n1*v+n2-1));
 wronganswer2:diff(g,v);</text>
    </questionvariables>
    <specificfeedback format="html">
      <text><![CDATA[<p>[[feedback:Result]]</p>]]></text>
    </specificfeedback>
    <questionnote format="moodle_auto_format">
      <text>\(\frac{dy}{dx}\) of \(y=\cosh^{-1}\left(\sqrt{{@n1@} {@v@} + {@n2@}}\right)\)  = \({@correctanswer@}\)</text>
    </questionnote>
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    <prtcorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:green;"><i class="fa fa-check"></i></span> Correct answer, well done.]]></text>
    </prtcorrect>
    <prtpartiallycorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:orange;"><i class="fa fa-adjust"></i></span> Your answer is partially correct.]]></text>
    </prtpartiallycorrect>
    <prtincorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:red;"><i class="fa fa-times"></i></span> Incorrect answer.]]></text>
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      <type>algebraic</type>
      <tans>correctanswer</tans>
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        <truefeedback format="html">
          <text><![CDATA[<p>To differentiate a function of a function, you must use the chain rule! It is \[\frac{df}{dg} \cdot\frac{dg}{dx}.\]</p>]]></text>
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      <node>
        <name>2</name>
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        <trueanswernote>Result-2-T </trueanswernote>
        <truefeedback format="html">
          <text><![CDATA[<p>You appear to have applied the chain rule incorrectly. It is \[\frac{df}{dg} \cdot\frac{dg}{dx}.\]</p>]]></text>
        </truefeedback>
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      <description/>
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        <name>ans1</name>
        <value>correctanswer</value>
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  </question>
</quiz>
