<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="stack">
    <name>
      <text>CR-Diff-09-misc-4.b</text>
    </name>
    <questiontext format="html">
      <text>Find the following derivative:
\[ \frac{d}{d{@v@}}[{@p@}]. \]

[[input:ans1]] [[validation:ans1]]</text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[To solve this question, you will have to use the chain rule <hint>calc_chain_rule</hint> more than once. 

Let \(f({@v@})={@p@}\) and \(g({@v@})={@pp@}\). Then, \(f({@v@})=e^{(g({@v@}))}\)

\[\begin{array}{rcl}\displaystyle 
 \frac{d}{d{@v@}}[{@p@}] &\displaystyle =&\displaystyle  \frac{df}{dg}\cdot \frac{dg}{dx} \\
   &\displaystyle =&\displaystyle  \frac{d}{dg}(e^{g})\cdot \frac{d}{dx}[{@pp@}]
 \end{array}\] 
 
To find \[\frac{d}{dx}[{@pp@}]\] you have to define a new function: \(h({@v@})={@ppp@}\), so that \(g({@v@})=h^{@rdm2@}\). So,

\[\begin{array}{rcl}\displaystyle 
 \frac{d}{d{@v@}}[{@pp@}] &\displaystyle =&\displaystyle  \frac{dg}{dh}\cdot \frac{dh}{d{@v@}} \\
   &\displaystyle =&\displaystyle  \frac{d}{dh}(h^{@rdm2@})\cdot \frac{d}{d{@v@}}({@ppp@}) \\
   &\displaystyle =&\displaystyle  {@rdm2@}h^{@rdm2-1@}\cdot ({@diff(ppp,v)@}) \\
   &\displaystyle =&\displaystyle  {@diff(pp,v)@}
 \end{array}\]  

So,

\[\begin{array}{rcl}\displaystyle 
 \frac{d}{d{@v@}}[{@p@}] &\displaystyle =&\displaystyle  \frac{d}{dg}(e^{g})\cdot \frac{d}{dx}[{@pp@}] \\
   &\displaystyle =&\displaystyle  e^g \cdot {@diff(pp,v)@} \\
   &\displaystyle =&\displaystyle  e^{@pp@}{@diff(pp,v)@} \\
   &\displaystyle =&\displaystyle  {@diff(p,v)@}
 \end{array}\]]]></text>
    </generalfeedback>
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    <penalty>0.1</penalty>
    <hidden>0</hidden>
    <idnumber/>
    <stackversion>
      <text/>
    </stackversion>
    <questionvariables>
      <text>v:x;
rdm1:2+rand(2);
rdm2:1/(2+rand(5));
ppp:v-v^rdm1;
pp:ppp^rdm2;
p:exp(pp);</text>
    </questionvariables>
    <specificfeedback format="html">
      <text><![CDATA[<p>[[feedback:Result]]</p>]]></text>
    </specificfeedback>
    <questionnote format="moodle_auto_format">
      <text>\[\frac{d}{d{@v@}}({@p@}) = {@diff(p,v)@} \]</text>
    </questionnote>
    <questiondescription format="moodle_auto_format">
      <text/>
    </questiondescription>
    <questionsimplify>1</questionsimplify>
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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>
    </prtincorrect>
    <decimals>.</decimals>
    <scientificnotation>*10</scientificnotation>
    <multiplicationsign>dot</multiplicationsign>
    <sqrtsign>1</sqrtsign>
    <complexno>i</complexno>
    <inversetrig>cos-1</inversetrig>
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    <input>
      <name>ans1</name>
      <type>algebraic</type>
      <tans>diff(p,v)</tans>
      <boxsize>15</boxsize>
      <strictsyntax>1</strictsyntax>
      <insertstars>0</insertstars>
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      <syntaxattribute>0</syntaxattribute>
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    </input>
    <prt>
      <name>Result</name>
      <value>1.0000000</value>
      <autosimplify>1</autosimplify>
      <feedbackstyle>1</feedbackstyle>
      <feedbackvariables>
        <text>q3:v+v^rdm1;
q2:q3^rdm2;
q:exp(q2);</text>
      </feedbackvariables>
      <node>
        <name>0</name>
        <description/>
        <answertest>Diff</answertest>
        <sans>ans1</sans>
        <tans>diff(p,v)</tans>
        <testoptions>v</testoptions>
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        <truescore>1.0000000</truescore>
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        <truenextnode>-1</truenextnode>
        <trueanswernote>Result-0-T </trueanswernote>
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          <text/>
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      <node>
        <name>1</name>
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        <answertest>AlgEquiv</answertest>
        <sans>ans1</sans>
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        <truenextnode>-1</truenextnode>
        <trueanswernote>Result-1-T </trueanswernote>
        <truefeedback format="html">
          <text>You appear to have lost the minus sign as your answer is the negative of what was expected.</text>
        </truefeedback>
        <falsescoremode>=</falsescoremode>
        <falsescore>0.0000000</falsescore>
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          <text/>
        </falsefeedback>
      </node>
      <node>
        <name>2</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>ans1</sans>
        <tans>diff(q,v)</tans>
        <testoptions/>
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        <trueanswernote>Result-2-T </trueanswernote>
        <truefeedback format="html">
          <text>You appear to have differentiated {@ppp@} incorrectly - there is a sign wrong.</text>
        </truefeedback>
        <falsescoremode>=</falsescoremode>
        <falsescore>0.0000000</falsescore>
        <falsepenalty/>
        <falsenextnode>3</falsenextnode>
        <falseanswernote>Result-2-F </falseanswernote>
        <falsefeedback format="html">
          <text/>
        </falsefeedback>
      </node>
      <node>
        <name>3</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>ans1</sans>
        <tans>diff(p,v)/diff(pp,v)</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>=</truescoremode>
        <truescore>0.0000000</truescore>
        <truepenalty/>
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        <trueanswernote>Result-3-T </trueanswernote>
        <truefeedback format="html">
          <text><![CDATA[You need to use the chain rule <hint>calc_chain_rule</hint> to differentiate {@p@} with respect to {@v@}.]]></text>
        </truefeedback>
        <falsescoremode>=</falsescoremode>
        <falsescore>0.0000000</falsescore>
        <falsepenalty/>
        <falsenextnode>-1</falsenextnode>
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          <text/>
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    <qtest>
      <testcase>1</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>ev(diff(p,v),simp)</value>
      </testinput>
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    <qtest>
      <testcase>2</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>ev(-diff(p,v),simp)</value>
      </testinput>
      <expected>
        <name>Result</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>Result-1-T </expectedanswernote>
      </expected>
    </qtest>
  </question>
</quiz>
