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<assessmentItem identifier="Q304071" title="derivative exp" adaptive="false" timeDependent="false" toolName="Testvision Online" toolVersion="50.0.11358" xmlns="http://www.imsglobal.org/xsd/imsqti_v2p1">
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      <value>14.88</value>
      <value>14.87;14.89</value>
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        <p>What is the derivative of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML" width=" 224"><semantics><mrow><mi>f</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>x</mi><mo stretchy="true" form="postfix">)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><msup><mi>e</mi><mrow><mn>3.6</mn><mi>x</mi></mrow></msup></mrow><annotation encoding="application/x-tex">f(x) = x^{2} e^{3.6x}</annotation></semantics></math>, evaluated at <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML" width=" 136"><semantics><mrow><mi>x</mi><mo>=</mo><mn>0.54</mn></mrow><annotation encoding="application/x-tex">x = 0.54</annotation></semantics></math>?</p>
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      <div class="rte_zone tveditor1">
        <p>Using the product rule for <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML" width=" 295"><semantics><mrow><mi>f</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>x</mi><mo stretchy="true" form="postfix">)</mo></mrow><mo>=</mo><mi>g</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>x</mi><mo stretchy="true" form="postfix">)</mo></mrow><mo>⋅</mo><mi>h</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>x</mi><mo stretchy="true" form="postfix">)</mo></mrow></mrow><annotation encoding="application/x-tex">f(x) = g(x) \cdot h(x)</annotation></semantics></math>, where <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML" width=" 163"><semantics><mrow><mi>g</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>x</mi><mo stretchy="true" form="postfix">)</mo></mrow><mo>:=</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">g(x) := x^{2}</annotation></semantics></math> and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML" width=" 193"><semantics><mrow><mi>h</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>x</mi><mo stretchy="true" form="postfix">)</mo></mrow><mo>:=</mo><msup><mi>e</mi><mrow><mn>3.6</mn><mi>x</mi></mrow></msup></mrow><annotation encoding="application/x-tex">h(x) := e^{3.6x}</annotation></semantics></math>, we obtain
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" width=" 857"><semantics><mtable><mtr><mtd columnalign="right"><mi>f</mi><mi>′</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>x</mi><mo stretchy="true" form="postfix">)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="right"><mrow><mo stretchy="true" form="prefix">[</mo><mi>g</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>x</mi><mo stretchy="true" form="postfix">)</mo></mrow><mo>⋅</mo><mi>h</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>x</mi><mo stretchy="true" form="postfix">)</mo></mrow><mo stretchy="true" form="postfix">]</mo></mrow><mi>′</mi><mo>=</mo><mi>g</mi><mi>′</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>x</mi><mo stretchy="true" form="postfix">)</mo></mrow><mo>⋅</mo><mi>h</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>x</mi><mo stretchy="true" form="postfix">)</mo></mrow><mo>+</mo><mi>g</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>x</mi><mo stretchy="true" form="postfix">)</mo></mrow><mo>⋅</mo><mi>h</mi><mi>′</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>x</mi><mo stretchy="true" form="postfix">)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="right"><mn>2</mn><msup><mi>x</mi><mrow><mn>2</mn><mo>−</mo><mn>1</mn></mrow></msup><mo>⋅</mo><msup><mi>e</mi><mrow><mn>3.6</mn><mi>x</mi></mrow></msup><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>⋅</mo><msup><mi>e</mi><mrow><mn>3.6</mn><mi>x</mi></mrow></msup><mo>⋅</mo><mn>3.6</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="right"><msup><mi>e</mi><mrow><mn>3.6</mn><mi>x</mi></mrow></msup><mo>⋅</mo><mrow><mo stretchy="true" form="prefix">(</mo><mn>2</mn><msup><mi>x</mi><mn>1</mn></msup><mo>+</mo><mn>3.6</mn><msup><mi>x</mi><mn>2</mn></msup><mo stretchy="true" form="postfix">)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="right"><msup><mi>e</mi><mrow><mn>3.6</mn><mi>x</mi></mrow></msup><mo>⋅</mo><msup><mi>x</mi><mn>1</mn></msup><mo>⋅</mo><mrow><mo stretchy="true" form="prefix">(</mo><mn>2</mn><mo>+</mo><mn>3.6</mn><mi>x</mi><mo stretchy="true" form="postfix">)</mo></mrow><mi>.</mi></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
f'(x) &amp;#38; = &amp;#38; [g(x) \cdot h(x)]' = g'(x) \cdot h(x) + g(x) \cdot h'(x) \\
      &amp;#38; = &amp;#38; 2 x^{2 - 1} \cdot e^{3.6x} + x^{2} \cdot e^{3.6x} \cdot 3.6 \\
      &amp;#38; = &amp;#38; e^{3.6x} \cdot(2 x^1 + 3.6 x^{2}) \\
      &amp;#38; = &amp;#38; e^{3.6x} \cdot x^1 \cdot (2 + 3.6x).\end{aligned}</annotation></semantics></math>
Evaluated at <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML" width=" 136"><semantics><mrow><mi>x</mi><mo>=</mo><mn>0.54</mn></mrow><annotation encoding="application/x-tex">x = 0.54</annotation></semantics></math>, the answer is
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" width=" 674"><semantics><mrow><msup><mi>e</mi><mrow><mn>3.6</mn><mo>⋅</mo><mn>0.54</mn></mrow></msup><mo>⋅</mo><msup><mn>0.54</mn><mn>1</mn></msup><mo>⋅</mo><mrow><mo stretchy="true" form="prefix">(</mo><mn>2</mn><mo>+</mo><mn>3.6</mn><mo>⋅</mo><mn>0.54</mn><mo stretchy="true" form="postfix">)</mo></mrow><mo>=</mo><mn>14.87987</mn><mi>.</mi></mrow><annotation encoding="application/x-tex">e^{3.6\cdot 0.54} \cdot 0.54^1 \cdot (2 + 3.6\cdot 0.54) = 14.87987.</annotation></semantics></math>
Thus, rounded to two digits we have <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML" width=" 263"><semantics><mrow><mi>f</mi><mi>′</mi><mrow><mo stretchy="true" form="prefix">(</mo><mn>0.54</mn><mo stretchy="true" form="postfix">)</mo></mrow><mo>=</mo><mn>14.88</mn></mrow><annotation encoding="application/x-tex">f'(0.54) = 14.88</annotation></semantics></math>.</p>
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    <div id="textBlockId_9963" class="textblock tvblock tvcss_1">
      <div class="rte_zone tveditor1">
        <p>Using the product rule for <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML" width=" 295"><semantics><mrow><mi>f</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>x</mi><mo stretchy="true" form="postfix">)</mo></mrow><mo>=</mo><mi>g</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>x</mi><mo stretchy="true" form="postfix">)</mo></mrow><mo>⋅</mo><mi>h</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>x</mi><mo stretchy="true" form="postfix">)</mo></mrow></mrow><annotation encoding="application/x-tex">f(x) = g(x) \cdot h(x)</annotation></semantics></math>, where <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML" width=" 163"><semantics><mrow><mi>g</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>x</mi><mo stretchy="true" form="postfix">)</mo></mrow><mo>:=</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">g(x) := x^{2}</annotation></semantics></math> and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML" width=" 193"><semantics><mrow><mi>h</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>x</mi><mo stretchy="true" form="postfix">)</mo></mrow><mo>:=</mo><msup><mi>e</mi><mrow><mn>3.6</mn><mi>x</mi></mrow></msup></mrow><annotation encoding="application/x-tex">h(x) := e^{3.6x}</annotation></semantics></math>, we obtain
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" width=" 857"><semantics><mtable><mtr><mtd columnalign="right"><mi>f</mi><mi>′</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>x</mi><mo stretchy="true" form="postfix">)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="right"><mrow><mo stretchy="true" form="prefix">[</mo><mi>g</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>x</mi><mo stretchy="true" form="postfix">)</mo></mrow><mo>⋅</mo><mi>h</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>x</mi><mo stretchy="true" form="postfix">)</mo></mrow><mo stretchy="true" form="postfix">]</mo></mrow><mi>′</mi><mo>=</mo><mi>g</mi><mi>′</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>x</mi><mo stretchy="true" form="postfix">)</mo></mrow><mo>⋅</mo><mi>h</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>x</mi><mo stretchy="true" form="postfix">)</mo></mrow><mo>+</mo><mi>g</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>x</mi><mo stretchy="true" form="postfix">)</mo></mrow><mo>⋅</mo><mi>h</mi><mi>′</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>x</mi><mo stretchy="true" form="postfix">)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="right"><mn>2</mn><msup><mi>x</mi><mrow><mn>2</mn><mo>−</mo><mn>1</mn></mrow></msup><mo>⋅</mo><msup><mi>e</mi><mrow><mn>3.6</mn><mi>x</mi></mrow></msup><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>⋅</mo><msup><mi>e</mi><mrow><mn>3.6</mn><mi>x</mi></mrow></msup><mo>⋅</mo><mn>3.6</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="right"><msup><mi>e</mi><mrow><mn>3.6</mn><mi>x</mi></mrow></msup><mo>⋅</mo><mrow><mo stretchy="true" form="prefix">(</mo><mn>2</mn><msup><mi>x</mi><mn>1</mn></msup><mo>+</mo><mn>3.6</mn><msup><mi>x</mi><mn>2</mn></msup><mo stretchy="true" form="postfix">)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="right"><msup><mi>e</mi><mrow><mn>3.6</mn><mi>x</mi></mrow></msup><mo>⋅</mo><msup><mi>x</mi><mn>1</mn></msup><mo>⋅</mo><mrow><mo stretchy="true" form="prefix">(</mo><mn>2</mn><mo>+</mo><mn>3.6</mn><mi>x</mi><mo stretchy="true" form="postfix">)</mo></mrow><mi>.</mi></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
f'(x) &amp;#38; = &amp;#38; [g(x) \cdot h(x)]' = g'(x) \cdot h(x) + g(x) \cdot h'(x) \\
      &amp;#38; = &amp;#38; 2 x^{2 - 1} \cdot e^{3.6x} + x^{2} \cdot e^{3.6x} \cdot 3.6 \\
      &amp;#38; = &amp;#38; e^{3.6x} \cdot(2 x^1 + 3.6 x^{2}) \\
      &amp;#38; = &amp;#38; e^{3.6x} \cdot x^1 \cdot (2 + 3.6x).\end{aligned}</annotation></semantics></math>
Evaluated at <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML" width=" 136"><semantics><mrow><mi>x</mi><mo>=</mo><mn>0.54</mn></mrow><annotation encoding="application/x-tex">x = 0.54</annotation></semantics></math>, the answer is
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" width=" 674"><semantics><mrow><msup><mi>e</mi><mrow><mn>3.6</mn><mo>⋅</mo><mn>0.54</mn></mrow></msup><mo>⋅</mo><msup><mn>0.54</mn><mn>1</mn></msup><mo>⋅</mo><mrow><mo stretchy="true" form="prefix">(</mo><mn>2</mn><mo>+</mo><mn>3.6</mn><mo>⋅</mo><mn>0.54</mn><mo stretchy="true" form="postfix">)</mo></mrow><mo>=</mo><mn>14.87987</mn><mi>.</mi></mrow><annotation encoding="application/x-tex">e^{3.6\cdot 0.54} \cdot 0.54^1 \cdot (2 + 3.6\cdot 0.54) = 14.87987.</annotation></semantics></math>
Thus, rounded to two digits we have <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML" width=" 263"><semantics><mrow><mi>f</mi><mi>′</mi><mrow><mo stretchy="true" form="prefix">(</mo><mn>0.54</mn><mo stretchy="true" form="postfix">)</mo></mrow><mo>=</mo><mn>14.88</mn></mrow><annotation encoding="application/x-tex">f'(0.54) = 14.88</annotation></semantics></math>.</p>
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