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<h3 class="hd hd-2">The Product Rule</h3>
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<h2 class="hd hd-2 unit-title">2. Product Rule</h2>
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<p><b class="bfseries">What functions can we differentiate so far?</b></p><p>
powers of [mathjaxinline]x[/mathjaxinline]<br/>polynomials<br/>sine and cosine <br/></p><p><b class="bfseries">Objectives</b></p><p>
At the end of this sequence, and after some practice, you should be able to: </p><ul class="itemize"><li><p>
Differentiate <span style="color:#27408C"><b class="bf">products</b></span> of functions. </p></li></ul><p><b class="bfseries">Contents: 12 pages</b></p><p>
6 videos (25 minutes 1x speed) 13 questions </p>
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<h2 class="hd hd-2 unit-title">3. What the derivative of a product is not</h2>
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Product of the derivatives?
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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The product
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(As above, suppose that [mathjaxinline]t[/mathjaxinline] is variable measuring time in seconds [mathjaxinline]s[/mathjaxinline], and [mathjaxinline]\ f[/mathjaxinline] and [mathjaxinline]g[/mathjaxinline] are functions of [mathjaxinline]t[/mathjaxinline] that output distances, measured in meters [mathjaxinline]m[/mathjaxinline].) </p>
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What units is [mathjaxinline]h(t) = f(t)g(t)[/mathjaxinline] measured in? </p>
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(Type [mathjaxinline]m[/mathjaxinline] for meters and [mathjaxinline]s[/mathjaxinline] for seconds. Type [mathjaxinline]*[/mathjaxinline] for multiplication; / for division; [mathjaxinline]\wedge[/mathjaxinline] for exponents.) </p>
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<th class="formulainput" scope="col">Example Entries</th>
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
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<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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The product&#39;s derivative
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What units is [mathjaxinline]h'(t)[/mathjaxinline] measured in? </p>
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(Type [mathjaxinline]m[/mathjaxinline] for meters and [mathjaxinline]s[/mathjaxinline] for seconds. Type [mathjaxinline]*[/mathjaxinline] for multiplication; / for division; [mathjaxinline]\wedge[/mathjaxinline] for exponents.) </p>
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<font color="#0078b0">2520</font>
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<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
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<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
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<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Do they match?
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Do the units of [mathjaxinline]h'(t)[/mathjaxinline] match the units of [mathjaxinline]\ f'(t)\cdot g'(t)[/mathjaxinline]? </p>
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<h2 class="hd hd-2 unit-title">5. Product of derivatives?</h2>
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Growing on the top
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At time [mathjaxinline]t=3[/mathjaxinline], we know the width of the rectangle is [mathjaxinline]\ f(3) = 50[/mathjaxinline] meters, and we said that the height is growing at [mathjaxinline]g'(3) = 2[/mathjaxinline] meters per second. How fast are we gaining area on the top of the rectangle? </p>
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<img alt="See text above." src="/assets/courseware/v1/5d1adbe6b256d8b259cf09b2366cf33f/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u2diff2_increasingareax.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
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\(\)
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What units is your answer in? </p>
<p>
(Type [mathjaxinline]m[/mathjaxinline] for meters and [mathjaxinline]s[/mathjaxinline] for seconds. Type [mathjaxinline]*[/mathjaxinline] for multiplication; / for division; [mathjaxinline]\wedge[/mathjaxinline] for exponents.) </p>
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<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
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<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
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<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
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<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
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<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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<h2 class="hd hd-2 unit-title">6. Area growth</h2>
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The width is 50 meters, and the top of the rectangle is moving up at 2 meters per second, so, on the top, we are gaining area at a rate of [mathjaxinline]50 \text {m} \times 2\ \text {m/s}[/mathjaxinline], or [mathjaxinline]100 \text {m^2/s}[/mathjaxinline]. The units work out perfectly!
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Growing on the right
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On the right, at time [mathjaxinline]t=3[/mathjaxinline], we have the height of the rectangle being [mathjaxinline]g(3) = 30[/mathjaxinline] meters, and we said that the width is growing at a rate of [mathjaxinline]\ f'(3) = 4[/mathjaxinline] meters per second. How fast are we gaining area on the right of the rectangle? </p>
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What units is your answer in? </p>
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(Type [mathjaxinline]m[/mathjaxinline] for meters and [mathjaxinline]s[/mathjaxinline] for seconds. Type [mathjaxinline]*[/mathjaxinline] for multiplication; / for division; [mathjaxinline]\wedge[/mathjaxinline] for exponents.) </p>
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<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
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<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
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<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
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<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
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<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
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<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
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<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Overall area growth
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Combining the two answers, what is the overall rate of growth of the area of the rectangle? </p>
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<h2 class="hd hd-2 unit-title">7. Product rule</h2>
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<h3 class="hd hd-2">Product rule</h3>
<div
id="video_diff_2-tab7-video1"
class="video closed"
data-metadata='{"lmsRootURL": "https://openlearninglibrary.mit.edu", "poster": null, "transcriptLanguages": {"en": "English"}, "showCaptions": "true", "ytMetadataEndpoint": "", "completionEnabled": false, "streams": "1.00:_a4_n-YZ_ag", "saveStateUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@diff_2-tab7-video1/handler/xmodule_handler/save_user_state", "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@diff_2-tab7-video1/handler/transcript/translation/__lang__", "savedVideoPosition": 0.0, "end": 0.0, "captionDataDir": null, "publishCompletionUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@diff_2-tab7-video1/handler/publish_completion", "generalSpeed": 1.0, "autoAdvance": false, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@diff_2-tab7-video1/handler/transcript/available_translations", "start": 0.0, "duration": 0.0, "prioritizeHls": false, "autohideHtml5": false, "saveStateEnabled": false, "autoplay": false, "transcriptLanguage": "en", "completionPercentage": 0.95, "recordedYoutubeIsAvailable": true, "speed": null, "sources": [], "ytTestTimeout": 1500, "ytApiUrl": "https://www.youtube.com/iframe_api"}'
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<h2 class="hd hd-2 unit-title">8. Product Rule</h2>
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<p>
If [mathjaxinline]h(x) = f(x)g(x)[/mathjaxinline], then </p><table id="a0000000211" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]h'(x) = f(x)g'(x)+ g(x)f'(x)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
at all points where the derivatives [mathjaxinline]\ f'(x)[/mathjaxinline] and [mathjaxinline]g'(x)[/mathjaxinline] are defined. </p>
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<h2 class="hd hd-2 unit-title">9. Some problems</h2>
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Product rule example
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<p>
Find the derivative of [mathjaxinline]j(x) = \sqrt {x} \cos x[/mathjaxinline]. </p>
<p>
(You can type cos(x), sin(x), and sqrt(x) if you need those functions. Type [mathjaxinline]*[/mathjaxinline] for multiplication; / for division; [mathjaxinline]\wedge[/mathjaxinline] for exponents.) </p>
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<th class="formulainput" scope="col">Example Entries</th>
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<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
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<font color="#0078b0">2/3</font>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
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<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
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<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
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<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
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<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Old McDonald
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<p>
Old McDonald has a farm. This summer he's been growing a very large watermelon - right now it weighs 100 pounds and is continuing to grow at a rate of 3 pounds per day. However, the market price of watermelon is currently 0.40 dollars per pound, and is decreasing at a rate of 0.01 dollars per pound per day. </p>
<p>
In dollars, what is the current value of Old McDonald's watermelon? </p>
<p>
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<span class="status unanswered" id="status_diff_2-tab9-problem2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_diff_2-tab9-problem2_2_1" class="answer"/>
<div id="input_diff_2-tab9-problem2_2_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
</p>
<p>
Let [mathjaxinline]w(d)[/mathjaxinline] denote the weight of the watermelon on day [mathjaxinline]d[/mathjaxinline], and [mathjaxinline]p(d)[/mathjaxinline] denote the market price per pound of watermelon on day [mathjaxinline]d[/mathjaxinline]. If [mathjaxinline]v(d)[/mathjaxinline] is the value of Old McDonald's watermelon on day [mathjaxinline]d[/mathjaxinline], express [mathjaxinline]v(d)[/mathjaxinline] in terms of [mathjaxinline]w(d)[/mathjaxinline] and [mathjaxinline]p(d)[/mathjaxinline]. </p>
<p>
(Type [mathjaxinline]*[/mathjaxinline] for multiplication; / for division; [mathjaxinline]\wedge[/mathjaxinline] for exponentiation.) </p>
<p>
<p style="display:inline">[mathjaxinline]v(d) =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_diff_2-tab9-problem2_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_diff_2-tab9-problem2_3_1" id="input_diff_2-tab9-problem2_3_1" aria-describedby="status_diff_2-tab9-problem2_3_1" value="" class="math"/>
<span class="trailing_text" id="trailing_text_diff_2-tab9-problem2_3_1"/>
<span class="status unanswered" id="status_diff_2-tab9-problem2_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_diff_2-tab9-problem2_3_1" class="answer"/>
<div id="display_diff_2-tab9-problem2_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_diff_2-tab9-problem2_3_1_dynamath" name="input_diff_2-tab9-problem2_3_1_dynamath"/>
</div>
</div></div>
</p>
<p>
In dollars per day, at what rate is the market value of Old McDonald's watermelon currently changing? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div id="formulaequationinput_diff_2-tab9-problem2_4_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_diff_2-tab9-problem2_4_1" id="input_diff_2-tab9-problem2_4_1" data-input-id="diff_2-tab9-problem2_4_1" value="" aria-describedby="status_diff_2-tab9-problem2_4_1" size="50"/>
<span class="trailing_text" id="trailing_text_diff_2-tab9-problem2_4_1"/>
<span class="status unanswered" id="status_diff_2-tab9-problem2_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_diff_2-tab9-problem2_4_1" class="answer"/>
<div id="input_diff_2-tab9-problem2_4_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_diff_2-tab9-problem2_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Old McDonald" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_diff_2-tab9-problem2" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_diff_2-tab9-problem2">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
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</div>
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is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
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</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="diff_2-tab9-problem2-problem-title">Answers are displayed within the problem
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data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
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</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="644e05caea7911efbdc412d4917dea95" data-runtime-version="1" data-block-type="vertical" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@vertical+block@diff_2-tab10">
<h2 class="hd hd-2 unit-title">10. Product Rule, Formally</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.1x+2T2019+type@video+block@diff_2-tab10-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="644e05caea7911efbdc412d4917dea95" data-runtime-version="1" data-block-type="video" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@video+block@diff_2-tab10-video1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">The Product Rule, Formally</h3>
<div
id="video_diff_2-tab10-video1"
class="video closed"
data-metadata='{"lmsRootURL": "https://openlearninglibrary.mit.edu", "poster": null, "transcriptLanguages": {"en": "English"}, "showCaptions": "true", "ytMetadataEndpoint": "", "completionEnabled": false, "streams": "1.00:YDPtfo6AZ0c", "saveStateUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@diff_2-tab10-video1/handler/xmodule_handler/save_user_state", "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@diff_2-tab10-video1/handler/transcript/translation/__lang__", "savedVideoPosition": 0.0, "end": 0.0, "captionDataDir": null, "publishCompletionUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@diff_2-tab10-video1/handler/publish_completion", "generalSpeed": 1.0, "autoAdvance": false, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@diff_2-tab10-video1/handler/transcript/available_translations", "start": 0.0, "duration": 0.0, "prioritizeHls": false, "autohideHtml5": false, "saveStateEnabled": false, "autoplay": false, "transcriptLanguage": "en", "completionPercentage": 0.95, "recordedYoutubeIsAvailable": true, "speed": null, "sources": [], "ytTestTimeout": 1500, "ytApiUrl": "https://www.youtube.com/iframe_api"}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="diff_2-tab10-video1"></div>
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<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_diff_2-tab10-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_diff_2-tab10-video1">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@diff_2-tab10-video1/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@diff_2-tab10-video1/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="644e05caea7911efbdc412d4917dea95" data-runtime-version="1" data-block-type="vertical" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@vertical+block@diff_2-tab11">
<h2 class="hd hd-2 unit-title">11. Basics</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@diff_2-tab11-problem1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="644e05caea7911efbdc412d4917dea95" data-runtime-version="1" data-block-type="problem" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@diff_2-tab11-problem1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_diff_2-tab11-problem1" class="problems-wrapper" role="group"
aria-labelledby="diff_2-tab11-problem1-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@diff_2-tab11-problem1" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@diff_2-tab11-problem1/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="diff_2-tab11-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@diff_2-tab11-problem1-problem-progress" tabindex="-1">
Example 1
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@diff_2-tab11-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
Do we need to use the product rule to differentiate [mathjaxinline]j(x) = \sqrt [3]{x} \cos x[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diff_2-tab11-problem1_2_1">
<fieldset aria-describedby="status_diff_2-tab11-problem1_2_1">
<div class="field">
<input type="radio" name="input_diff_2-tab11-problem1_2_1" id="input_diff_2-tab11-problem1_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="diff_2-tab11-problem1_2_1-choice_1-label" for="input_diff_2-tab11-problem1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_diff_2-tab11-problem1_2_1">
<text> Yes</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_2-tab11-problem1_2_1" id="input_diff_2-tab11-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diff_2-tab11-problem1_2_1-choice_2-label" for="input_diff_2-tab11-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diff_2-tab11-problem1_2_1">
<text> No</text>
</label>
</div>
<span id="answer_diff_2-tab11-problem1_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_diff_2-tab11-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_diff_2-tab11-problem1_solution_1"/>
</div></p>
<script src="/assets/courseware/v1/14976dbaebca69b9b0e6989fe9838bb1/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/st.js" type="text/javascript"/>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Example 1" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_diff_2-tab11-problem1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_diff_2-tab11-problem1">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
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is-hidden"
tabindex="-1">
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</span>
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<span class="notification-message" aria-describedby="diff_2-tab11-problem1-problem-title">None
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<span class="notification-message" aria-describedby="diff_2-tab11-problem1-problem-title">Answers are displayed within the problem
</span>
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</div>
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"
data-graded="True">
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<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
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</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@diff_2-tab11-problem2">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="644e05caea7911efbdc412d4917dea95" data-runtime-version="1" data-block-type="problem" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@diff_2-tab11-problem2">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_diff_2-tab11-problem2" class="problems-wrapper" role="group"
aria-labelledby="diff_2-tab11-problem2-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@diff_2-tab11-problem2" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@diff_2-tab11-problem2/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="diff_2-tab11-problem2-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@diff_2-tab11-problem2-problem-progress" tabindex="-1">
Example 2
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@diff_2-tab11-problem2-problem-progress"></div>
<div class="problem">
<div>
<p>
Do we need to use the product rule to differentiate [mathjaxinline]k(x) = \pi x^5[/mathjaxinline]? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diff_2-tab11-problem2_2_1">
<fieldset aria-describedby="status_diff_2-tab11-problem2_2_1">
<div class="field">
<input type="radio" name="input_diff_2-tab11-problem2_2_1" id="input_diff_2-tab11-problem2_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="diff_2-tab11-problem2_2_1-choice_1-label" for="input_diff_2-tab11-problem2_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_diff_2-tab11-problem2_2_1"> <text> Yes</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_2-tab11-problem2_2_1" id="input_diff_2-tab11-problem2_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diff_2-tab11-problem2_2_1-choice_2-label" for="input_diff_2-tab11-problem2_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diff_2-tab11-problem2_2_1"> <text> No</text>
</label>
</div>
<span id="answer_diff_2-tab11-problem2_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_diff_2-tab11-problem2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div> </p>
<p>
<div class="solution-span">
<span id="solution_diff_2-tab11-problem2_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Example 2" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_diff_2-tab11-problem2" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_diff_2-tab11-problem2">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
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is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
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<div class="notification warning notification-save
is-hidden"
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<span class="icon fa fa-save" aria-hidden="true"></span>
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</div>
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is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="diff_2-tab11-problem2-problem-title">Answers are displayed within the problem
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<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
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"
data-graded="True">
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<span class="sr">Loading…</span>
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</div>
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Example 3
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Do we need to use the product rule to differentiate [mathjaxinline]p(x) = \sin (60 x)[/mathjaxinline]? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diff_2-tab11-problem3_2_1">
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When can we use the product rule?
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Suppose we know that [mathjaxinline]\ f(10) = 1776[/mathjaxinline], [mathjaxinline]g(10) = \pi[/mathjaxinline], and [mathjaxinline]h(10) = 1776\pi[/mathjaxinline], and that all of these functions are differentiable. Can we conclude that </p>
<table cellpadding="7" cellspacing="0" class="equation" id="a0000000213" style="table-layout:auto" width="100%">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]h'(10) = 1776 g'(10) + \pi f'(10)?[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<h2 class="hd hd-2 unit-title">12. Practice product rule</h2>
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<h2 class="hd hd-2 unit-title">13. Summary</h2>
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<p><b class="bfseries">The product rule</b></p><p>
If [mathjaxinline]h(x) = f(x)g(x)[/mathjaxinline], then </p><table id="a0000000214" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]h'(x) = f(x)g'(x)+ g(x)f'(x)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
at all points where the derivatives [mathjaxinline]\ f'(x)[/mathjaxinline] and [mathjaxinline]g'(x)[/mathjaxinline] are defined. </p>
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