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<h2 class="hd hd-2 unit-title">1. Motivation</h2>
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<h3 class="hd hd-2">The chain rule</h3>
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<h2 class="hd hd-2 unit-title">2. The chain 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/>trigonometric functions <br/>sums of basic functions <br/>products of basic functions<br/>quotients of basic functions <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:#99182C"><b class="bf">compositions</b></span> of functions. </p></li></ul><p><b class="bfseries">Contents: 13 pages</b></p><p>
9 videos (42 minutes 1x speed) 20 questions </p>
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<h2 class="hd hd-2 unit-title">3. Changing Units</h2>
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Changing Units
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We often want to adjust a function by changing the units on the function's inputs or outputs. For instance, we might want to measure time in minutes as opposed to seconds. In these next few problems, we'll see how that affects derivatives. </p>
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Suppose that the variable [mathjaxinline]x[/mathjaxinline] measures the position of a bicycle along a road, in meters, and that [mathjaxinline]t[/mathjaxinline] measures the amount of time since the start of the journey in minutes and [mathjaxinline]u[/mathjaxinline] measures that same time in seconds. </p>
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What is the relationship between [mathjaxinline]u[/mathjaxinline] and [mathjaxinline]t[/mathjaxinline]? </p>
<p>
(Type [mathjaxinline]*[/mathjaxinline] for multiplication. Type / for division.) </p>
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<p style="display:inline">[mathjaxinline]u =[/mathjaxinline] </p>
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Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
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Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Derivatives
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The position [mathjaxinline]x[/mathjaxinline] is a function of time. We can write it as a function of either [mathjaxinline]u[/mathjaxinline] or [mathjaxinline]t[/mathjaxinline], since both are time variables. To distinguish, let's say that [mathjaxinline]x = f(u)[/mathjaxinline] and that [mathjaxinline]x = g(t)[/mathjaxinline]. </p>
<p>
Suppose that [mathjaxinline]\ f(u) = 2u + 100[/mathjaxinline]. What is [mathjaxinline]\displaystyle \frac{dx}{du}[/mathjaxinline]? </p>
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What units is [mathjaxinline]\displaystyle \frac{dx}{du}[/mathjaxinline] measured in? </p>
<p>
(Type [mathjaxinline]m[/mathjaxinline] for meters, [mathjaxinline]min[/mathjaxinline] for minutes, or [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>
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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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Scaling factor
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Given our formula for [mathjaxinline]x[/mathjaxinline] in terms of [mathjaxinline]u[/mathjaxinline], what is the formula for [mathjaxinline]x[/mathjaxinline] in terms of [mathjaxinline]t[/mathjaxinline]? </p>
<p>
(Type [mathjaxinline]*[/mathjaxinline] for multiplication; e.g. 2[mathjaxinline]*[/mathjaxinline]t for [mathjaxinline]2t[/mathjaxinline]. Type / for division; e.g. 2/t for [mathjaxinline]\displaystyle \frac{2}{t}[/mathjaxinline]. Type [mathjaxinline]\wedge[/mathjaxinline] for exponents; e.g. t[mathjaxinline]\wedge[/mathjaxinline]2 for [mathjaxinline]t^2[/mathjaxinline].) </p>
<p>
<p style="display:inline">[mathjaxinline]x = g(t) =[/mathjaxinline] </p>
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What is [mathjaxinline]\displaystyle \frac{dx}{dt}[/mathjaxinline]? </p>
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What units is [mathjaxinline]\displaystyle \frac{dx}{dt}[/mathjaxinline] measured in? Type [mathjaxinline]m[/mathjaxinline] for meters, [mathjaxinline]min[/mathjaxinline] for minutes, or [mathjaxinline]s[/mathjaxinline] for seconds. </p>
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<div class="formulainput">
<table class="formulainput">
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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>
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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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<h2 class="hd hd-2 unit-title">4. Another viewpoint</h2>
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<p>
Let's see another way to think about this. Suppose we have a function which is a composition of two other functions, [mathjaxinline]h(x) = f\left(g(x)\right)[/mathjaxinline], and we want to think about [mathjaxinline]h'(2)[/mathjaxinline]. Suppose that </p><table id="a0000000240" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000241"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle g(2)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = 9 \quad \text {and}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.3)</td></tr><tr id="a0000000242"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle f(9)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = 5, \quad \text {so}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.4)</td></tr><tr id="a0000000243"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle h(2)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = 5.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.5)</td></tr></table><p>
Furthermore, we'll say that the 2 refers to 2 seconds, the 9 refers to 9 meters, and the 5 refers to 5 kilograms. </p><p>
(<i class="itshape">Use these data in all of the problems that follow on this page.</i>) </p>
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Let's suppose that [mathjaxinline]g'(2) = 3[/mathjaxinline] meters per second. What is your best estimate for [mathjaxinline]g(2.01)[/mathjaxinline]? </p>
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Changing the input 2
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Now let's suppose that [mathjaxinline]\ f'(9) = 4[/mathjaxinline] kilograms per meter. What is your best estimate for [mathjaxinline]\ f(g(2.01))[/mathjaxinline]? </p>
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Putting it together
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Note that you just estimated [mathjaxinline]\ f\left(g(2.01)\right)[/mathjaxinline], which is the same as [mathjaxinline]h(2.01)[/mathjaxinline]. We know that [mathjaxinline]h(2) = 5[/mathjaxinline], so we know approximately how much [mathjaxinline]h[/mathjaxinline] has changed as our input moved from [mathjaxinline]2[/mathjaxinline] to [mathjaxinline]2.01[/mathjaxinline]. Given this, approximately what do you think [mathjaxinline]h'(2)[/mathjaxinline] should be? </p>
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<span class="trailing_text" id="trailing_text_diff_4-tab4-problem3_2_1"> kg/sec</span>
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Three link chain
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<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="c8a54462ea7811efbdc412d4917dea95" data-runtime-version="1" data-block-type="html" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@diff_4-tab7-text1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>
If [mathjaxinline]h(x) = f(g(x))[/mathjaxinline], then </p><table id="a0000000244" 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'\left(g(x)\right) g'(x)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
at all points where the derivatives [mathjaxinline]\ f'(g(x))[/mathjaxinline] and [mathjaxinline]g'(x)[/mathjaxinline] are defined. </p><p>
Alternatively, if [mathjaxinline]y = f(u)[/mathjaxinline], and [mathjaxinline]u=g(x)[/mathjaxinline], then </p><table id="a0000000245" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\left. \frac{dy}{dx} \right|_{x = a} = \left. \frac{dy}{du} \right|_{u=g(a)} \left. \frac{du}{dx} \right|_{x=a}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
at any point [mathjaxinline]x=a[/mathjaxinline] where the derivatives on the right hand side are defined. </p>
</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="c8a54462ea7811efbdc412d4917dea95" 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_4-tab8">
<h2 class="hd hd-2 unit-title">8. Example 1</h2>
<div class="vert-mod">
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{"xmodule-type": "Problem"}
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<div id="problem_diff_4-tab8-problem1" class="problems-wrapper" role="group"
aria-labelledby="diff_4-tab8-problem1-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@diff_4-tab8-problem1" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@diff_4-tab8-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_4-tab8-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@diff_4-tab8-problem1-problem-progress" tabindex="-1">
Inner function
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@diff_4-tab8-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
We want to differentiate [mathjaxinline]p(x) = \cos ^3 x[/mathjaxinline]. We can use the chain rule if we can recognize [mathjaxinline]p(x)[/mathjaxinline] as a composition [mathjaxinline]\ f(g(x))[/mathjaxinline]. Which of the following functions should take the role of the inner function [mathjaxinline]g(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_4-tab8-problem1_2_1">
<fieldset aria-describedby="status_diff_4-tab8-problem1_2_1">
<div class="field">
<input type="radio" name="input_diff_4-tab8-problem1_2_1" id="input_diff_4-tab8-problem1_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="diff_4-tab8-problem1_2_1-choice_1-label" for="input_diff_4-tab8-problem1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab8-problem1_2_1">
<text> [mathjaxinline]x[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_4-tab8-problem1_2_1" id="input_diff_4-tab8-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diff_4-tab8-problem1_2_1-choice_2-label" for="input_diff_4-tab8-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab8-problem1_2_1">
<text> [mathjaxinline]\cos x[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_4-tab8-problem1_2_1" id="input_diff_4-tab8-problem1_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="diff_4-tab8-problem1_2_1-choice_3-label" for="input_diff_4-tab8-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab8-problem1_2_1">
<text> [mathjaxinline]x^3[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_4-tab8-problem1_2_1" id="input_diff_4-tab8-problem1_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="diff_4-tab8-problem1_2_1-choice_4-label" for="input_diff_4-tab8-problem1_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab8-problem1_2_1">
<text> [mathjaxinline]\cos ^3 x[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_4-tab8-problem1_2_1" id="input_diff_4-tab8-problem1_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="diff_4-tab8-problem1_2_1-choice_5-label" for="input_diff_4-tab8-problem1_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab8-problem1_2_1">
<text> None of these</text>
</label>
</div>
<span id="answer_diff_4-tab8-problem1_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_diff_4-tab8-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_4-tab8-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="Inner function" />
<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_4-tab8-problem1" >
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<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
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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>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.01.1x+2T2019+type@video+block@diff_4-tab8-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="c8a54462ea7811efbdc412d4917dea95" 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_4-tab8-video1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
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<h3 class="hd hd-2">Using the Chain Rule</h3>
<div
id="video_diff_4-tab8-video1"
class="video closed"
data-metadata='{"lmsRootURL": "https://openlearninglibrary.mit.edu", "poster": null, "transcriptLanguages": {"en": "English"}, "showCaptions": "true", "ytMetadataEndpoint": "", "completionEnabled": false, "streams": "1.00:ts3qbYHIFu0", "saveStateUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@diff_4-tab8-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_4-tab8-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_4-tab8-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_4-tab8-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'
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tabindex="-1"
>
<div class="focus_grabber first"></div>
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Your browser does not support this video format. Try using a different browser.
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<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
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<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_diff_4-tab8-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_diff_4-tab8-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_4-tab8-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_4-tab8-video1/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
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<div class="vert vert-2" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@diff_4-tab8-problem2">
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aria-labelledby="diff_4-tab8-problem2-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@diff_4-tab8-problem2" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@diff_4-tab8-problem2/handler/xmodule_handler"
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<h3 class="hd hd-3 problem-header" id="diff_4-tab8-problem2-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@diff_4-tab8-problem2-problem-progress" tabindex="-1">
Example 2
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@diff_4-tab8-problem2-problem-progress"></div>
<div class="problem">
<div>
<p>
Let [mathjaxinline]\ f(x) = \sqrt {x^3 + 2x + 1}[/mathjaxinline]. Calculate [mathjaxinline]\ f'(1)[/mathjaxinline]. </p>
<p>
(You can type sqrt(x) for [mathjaxinline]\sqrt {x}[/mathjaxinline].) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_diff_4-tab8-problem2_2_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_diff_4-tab8-problem2_2_1" id="input_diff_4-tab8-problem2_2_1" data-input-id="diff_4-tab8-problem2_2_1" value="" aria-describedby="status_diff_4-tab8-problem2_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_diff_4-tab8-problem2_2_1"/>
<span class="status unanswered" id="status_diff_4-tab8-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_4-tab8-problem2_2_1" class="answer"/>
<div id="input_diff_4-tab8-problem2_2_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
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<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
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<span>
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<h4>Formula Input Guide</h4>
<a class="mymodal-btn-close" href="#">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<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 \(\lambda \)
</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 \( (2\pi+y)dy \)
</td>
</tr>
</tbody>
</table>
</div>
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<a class="mymodal-btn" href="#">Close Guide</a>
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Let [mathjaxinline]g(x) = \sin (x^2 + 2x)[/mathjaxinline]. Calculate [mathjaxinline]g'(x)[/mathjaxinline]. </p>
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(You can type sin and cos for the sine and cosine functions. Type [mathjaxinline]*[/mathjaxinline] for multiplication; / for division; [mathjaxinline]\wedge[/mathjaxinline] for exponents.) </p>
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<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>
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<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">9. More complicated functions</h2>
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What happens last?
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@diff_4-tab9-problem1-problem-progress"></div>
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<p>
Let's think about differentiating a similar function, [mathjaxinline]g(x) = \sqrt {x \tan x}[/mathjaxinline]. </p>
<p>
First, we analyze how to calculate [mathjaxinline]g(x)[/mathjaxinline] itself. What's the last step we take? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diff_4-tab9-problem1_2_1">
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<text> Multiply two simpler functions</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_4-tab9-problem1_2_1" id="input_diff_4-tab9-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diff_4-tab9-problem1_2_1-choice_2-label" for="input_diff_4-tab9-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab9-problem1_2_1">
<text> Compose with the square root function</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_4-tab9-problem1_2_1" id="input_diff_4-tab9-problem1_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="diff_4-tab9-problem1_2_1-choice_3-label" for="input_diff_4-tab9-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab9-problem1_2_1">
<text> Compose with the tangent function</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_4-tab9-problem1_2_1" id="input_diff_4-tab9-problem1_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="diff_4-tab9-problem1_2_1-choice_4-label" for="input_diff_4-tab9-problem1_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab9-problem1_2_1">
<text> None of these</text>
</label>
</div>
<span id="answer_diff_4-tab9-problem1_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_diff_4-tab9-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>
Given the answer to the previous question, what differentiation rule will we need to apply first in order to differentiate [mathjaxinline]g[/mathjaxinline]? </p>
<p>
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<text> Product rule</text>
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<input type="radio" name="input_diff_4-tab9-problem1_3_1" id="input_diff_4-tab9-problem1_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diff_4-tab9-problem1_3_1-choice_2-label" for="input_diff_4-tab9-problem1_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab9-problem1_3_1">
<text> Quotient Rule</text>
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<input type="radio" name="input_diff_4-tab9-problem1_3_1" id="input_diff_4-tab9-problem1_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="diff_4-tab9-problem1_3_1-choice_3-label" for="input_diff_4-tab9-problem1_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab9-problem1_3_1">
<text> Chain Rule</text>
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<h3 class="hd hd-2">Differentiating Another Complicated Function</h3>
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<h2 class="hd hd-2 unit-title">10. Exercises</h2>
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What happens last 2?
</h3>
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<p>
As mentioned in the previous video, in order to determine how to differentiate a complicated function, we identify the last step needed to evaluate the function. If the last step is an arithmetic operation, such as taking a product or quotient, then we apply the product rule or quotient rule. If the last step is to compose with a &#8220;basic function," such as a square root or sine function, then we apply the chain rule. </p>
<p>
Determine the last step when evaluating the following functions. (Note you are evaluating the function, not a derivative in these questions!) </p>
<p>
[mathjaxinline]\displaystyle f(x) = \sin \left( \frac{x}{x^2 + 1} \right)[/mathjaxinline] <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diff_4-tab10-problem1_2_1">
<fieldset aria-describedby="status_diff_4-tab10-problem1_2_1">
<div class="field">
<input type="radio" name="input_diff_4-tab10-problem1_2_1" id="input_diff_4-tab10-problem1_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="diff_4-tab10-problem1_2_1-choice_1-label" for="input_diff_4-tab10-problem1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab10-problem1_2_1"> <text> Take a product</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_4-tab10-problem1_2_1" id="input_diff_4-tab10-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diff_4-tab10-problem1_2_1-choice_2-label" for="input_diff_4-tab10-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab10-problem1_2_1"> <text> Take a quotient</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_4-tab10-problem1_2_1" id="input_diff_4-tab10-problem1_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="diff_4-tab10-problem1_2_1-choice_3-label" for="input_diff_4-tab10-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab10-problem1_2_1"> <text> Apply a basic function</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_4-tab10-problem1_2_1" id="input_diff_4-tab10-problem1_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="diff_4-tab10-problem1_2_1-choice_4-label" for="input_diff_4-tab10-problem1_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab10-problem1_2_1"> <text> None of these</text>
</label>
</div>
<span id="answer_diff_4-tab10-problem1_2_1"/>
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<span class="status unanswered" id="status_diff_4-tab10-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>
[mathjaxinline]g(x) = \sqrt [3]{\sin (x^3) }[/mathjaxinline] <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diff_4-tab10-problem1_3_1">
<fieldset aria-describedby="status_diff_4-tab10-problem1_3_1">
<div class="field">
<input type="radio" name="input_diff_4-tab10-problem1_3_1" id="input_diff_4-tab10-problem1_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="diff_4-tab10-problem1_3_1-choice_1-label" for="input_diff_4-tab10-problem1_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab10-problem1_3_1"> <text> Take a product</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_4-tab10-problem1_3_1" id="input_diff_4-tab10-problem1_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diff_4-tab10-problem1_3_1-choice_2-label" for="input_diff_4-tab10-problem1_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab10-problem1_3_1"> <text> Take a quotient</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_4-tab10-problem1_3_1" id="input_diff_4-tab10-problem1_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="diff_4-tab10-problem1_3_1-choice_3-label" for="input_diff_4-tab10-problem1_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab10-problem1_3_1"> <text> Apply a basic function</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_4-tab10-problem1_3_1" id="input_diff_4-tab10-problem1_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="diff_4-tab10-problem1_3_1-choice_4-label" for="input_diff_4-tab10-problem1_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab10-problem1_3_1"> <text> None of these</text>
</label>
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<span id="answer_diff_4-tab10-problem1_3_1"/>
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<span class="status unanswered" id="status_diff_4-tab10-problem1_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
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</div></div> </p>
<p>
[mathjaxinline]j(x) = (x + \sqrt {x})^{100}[/mathjaxinline] <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diff_4-tab10-problem1_4_1">
<fieldset aria-describedby="status_diff_4-tab10-problem1_4_1">
<div class="field">
<input type="radio" name="input_diff_4-tab10-problem1_4_1" id="input_diff_4-tab10-problem1_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="diff_4-tab10-problem1_4_1-choice_1-label" for="input_diff_4-tab10-problem1_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab10-problem1_4_1"> <text> Take a product</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_4-tab10-problem1_4_1" id="input_diff_4-tab10-problem1_4_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diff_4-tab10-problem1_4_1-choice_2-label" for="input_diff_4-tab10-problem1_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab10-problem1_4_1"> <text> Take a quotient</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_4-tab10-problem1_4_1" id="input_diff_4-tab10-problem1_4_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="diff_4-tab10-problem1_4_1-choice_3-label" for="input_diff_4-tab10-problem1_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab10-problem1_4_1"> <text> Apply a basic function</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_4-tab10-problem1_4_1" id="input_diff_4-tab10-problem1_4_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="diff_4-tab10-problem1_4_1-choice_4-label" for="input_diff_4-tab10-problem1_4_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab10-problem1_4_1"> <text> None of these</text>
</label>
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<span id="answer_diff_4-tab10-problem1_4_1"/>
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<span class="status unanswered" id="status_diff_4-tab10-problem1_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
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</div></div> </p>
<p>
[mathjaxinline]k(x) = \cos x \, \sin ^4 x[/mathjaxinline] <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diff_4-tab10-problem1_5_1">
<fieldset aria-describedby="status_diff_4-tab10-problem1_5_1">
<div class="field">
<input type="radio" name="input_diff_4-tab10-problem1_5_1" id="input_diff_4-tab10-problem1_5_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="diff_4-tab10-problem1_5_1-choice_1-label" for="input_diff_4-tab10-problem1_5_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab10-problem1_5_1"> <text> Take a product</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_4-tab10-problem1_5_1" id="input_diff_4-tab10-problem1_5_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diff_4-tab10-problem1_5_1-choice_2-label" for="input_diff_4-tab10-problem1_5_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab10-problem1_5_1"> <text> Take a quotient</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_4-tab10-problem1_5_1" id="input_diff_4-tab10-problem1_5_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="diff_4-tab10-problem1_5_1-choice_3-label" for="input_diff_4-tab10-problem1_5_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab10-problem1_5_1"> <text> Apply a basic function</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_4-tab10-problem1_5_1" id="input_diff_4-tab10-problem1_5_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="diff_4-tab10-problem1_5_1-choice_4-label" for="input_diff_4-tab10-problem1_5_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab10-problem1_5_1"> <text> None of these</text>
</label>
</div>
<span id="answer_diff_4-tab10-problem1_5_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_diff_4-tab10-problem1_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div> </p>
<p>
[mathjaxinline]p(x) = \sqrt {2x+1}[/mathjaxinline] <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 5" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diff_4-tab10-problem1_6_1">
<fieldset aria-describedby="status_diff_4-tab10-problem1_6_1">
<div class="field">
<input type="radio" name="input_diff_4-tab10-problem1_6_1" id="input_diff_4-tab10-problem1_6_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="diff_4-tab10-problem1_6_1-choice_1-label" for="input_diff_4-tab10-problem1_6_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab10-problem1_6_1"> <text> Take a product</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_4-tab10-problem1_6_1" id="input_diff_4-tab10-problem1_6_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diff_4-tab10-problem1_6_1-choice_2-label" for="input_diff_4-tab10-problem1_6_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab10-problem1_6_1"> <text> Take a quotient</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_4-tab10-problem1_6_1" id="input_diff_4-tab10-problem1_6_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="diff_4-tab10-problem1_6_1-choice_3-label" for="input_diff_4-tab10-problem1_6_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab10-problem1_6_1"> <text> Apply a basic function</text>
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<input type="radio" name="input_diff_4-tab10-problem1_6_1" id="input_diff_4-tab10-problem1_6_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="diff_4-tab10-problem1_6_1-choice_4-label" for="input_diff_4-tab10-problem1_6_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab10-problem1_6_1"> <text> None of these</text>
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<p>
[mathjaxinline]q(x) = \displaystyle {\tan \left(3x^2 + \sin (17x)\right) \, \left( \frac{\cos x}{x} \right)^{13}}[/mathjaxinline] <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 6" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diff_4-tab10-problem1_7_1">
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<input type="radio" name="input_diff_4-tab10-problem1_7_1" id="input_diff_4-tab10-problem1_7_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="diff_4-tab10-problem1_7_1-choice_1-label" for="input_diff_4-tab10-problem1_7_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab10-problem1_7_1"> <text> Take a product</text>
</label>
</div>
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<input type="radio" name="input_diff_4-tab10-problem1_7_1" id="input_diff_4-tab10-problem1_7_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diff_4-tab10-problem1_7_1-choice_2-label" for="input_diff_4-tab10-problem1_7_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab10-problem1_7_1"> <text> Take a quotient</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_4-tab10-problem1_7_1" id="input_diff_4-tab10-problem1_7_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="diff_4-tab10-problem1_7_1-choice_3-label" for="input_diff_4-tab10-problem1_7_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab10-problem1_7_1"> <text> Apply a basic function</text>
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<input type="radio" name="input_diff_4-tab10-problem1_7_1" id="input_diff_4-tab10-problem1_7_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="diff_4-tab10-problem1_7_1-choice_4-label" for="input_diff_4-tab10-problem1_7_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab10-problem1_7_1"> <text> None of these</text>
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<h3 class="hd hd-3 problem-header" id="diff_4-tab10-problem2-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@diff_4-tab10-problem2-problem-progress" tabindex="-1">
Example 6
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<div class="problem">
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<p>
Let [mathjaxinline]\ \displaystyle f(x) = \sin \left( \frac{x}{x^2 + 1} \right)[/mathjaxinline]. Calculate [mathjaxinline]\ f'(x)[/mathjaxinline]. </p>
<p>
(You can type sin(x) or cos(x) for those functions. Type [mathjaxinline]*[/mathjaxinline] for multiplication; / for division; [mathjaxinline]\wedge[/mathjaxinline] for exponents. Use parentheses ( ) to designate order of operations.) </p>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_diff_4-tab10-problem2_2_1" class="inputtype formulaequationinput">
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\(\)
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<div class="formulainput">
<table class="formulainput">
<tbody>
<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 \(\lambda \)
</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 \( (2\pi+y)dy \)
</td>
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Example 7
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Let [mathjaxinline]k(x) = \cos x \, \sin ^4 x[/mathjaxinline]. Calculate [mathjaxinline]k'(x)[/mathjaxinline]. </p>
<p>
(You can type sin(x) or cos(x) for those functions. Type [mathjaxinline]*[/mathjaxinline] for multiplication; / for division; [mathjaxinline]\wedge[/mathjaxinline] for exponents. Use parentheses ( ) to designate order of operations; e.g. type <b class="bf">(sin(x))^2</b> for [mathjaxinline]\sin ^2(x)[/mathjaxinline]. ) </p>
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\(\)
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<div class="formulainput">
<table class="formulainput">
<tbody>
<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 \(\lambda \)
</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 \( (2\pi+y)dy \)
</td>
</tr>
</tbody>
</table>
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<h2 class="hd hd-2 unit-title">11. Quotient Rule vs Product and Chain Rule</h2>
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Which rule?
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<p>
To differentiate the function [mathjaxinline]\displaystyle {h(x) = \frac{1}{x^2 + x}}[/mathjaxinline], do we need the Chain Rule or the Quotient Rule? </p>
<p>
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<text> We have to use the Chain Rule</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_4-tab11-problem1_2_1" id="input_diff_4-tab11-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diff_4-tab11-problem1_2_1-choice_2-label" for="input_diff_4-tab11-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab11-problem1_2_1">
<text> We have to use the Quotient Rule</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_4-tab11-problem1_2_1" id="input_diff_4-tab11-problem1_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="diff_4-tab11-problem1_2_1-choice_3-label" for="input_diff_4-tab11-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab11-problem1_2_1">
<text> We can use either rule</text>
</label>
</div>
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<input type="radio" name="input_diff_4-tab11-problem1_2_1" id="input_diff_4-tab11-problem1_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="diff_4-tab11-problem1_2_1-choice_4-label" for="input_diff_4-tab11-problem1_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab11-problem1_2_1">
<text> We don't need to use either of them</text>
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<h3 class="hd hd-2">Chain Rule vs. Quotient Rule</h3>
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<h2 class="hd hd-2 unit-title">12. Review Problems</h2>
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Review 1
</h3>
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<div class="problem">
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<p>
True or false: If [mathjaxinline]g(x) = f(x)^2[/mathjaxinline], then [mathjaxinline]g'(x) = 2f(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_4-tab12-problem1_2_1">
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<text> True</text>
</label>
</div>
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<input type="radio" name="input_diff_4-tab12-problem1_2_1" id="input_diff_4-tab12-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diff_4-tab12-problem1_2_1-choice_2-label" for="input_diff_4-tab12-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diff_4-tab12-problem1_2_1">
<text> False</text>
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Suppose we know the following about functions [mathjaxinline]\ f[/mathjaxinline] and [mathjaxinline]g[/mathjaxinline]: </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000255" style="table-layout:auto" width="100%">
<tr id="a0000000256">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle f(3)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle = 5[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \qquad f'(3)[/mathjaxinline]
</td>
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[mathjaxinline]\displaystyle = 3[/mathjaxinline]
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<tr id="a0000000257">
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<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle f(4)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle = 2[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \qquad f'(4)[/mathjaxinline]
</td>
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[mathjaxinline]\displaystyle = -3[/mathjaxinline]
</td>
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<tr id="a0000000258">
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[mathjaxinline]\displaystyle f(5)[/mathjaxinline]
</td>
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[mathjaxinline]\displaystyle = 1[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \qquad f'(5)[/mathjaxinline]
</td>
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[mathjaxinline]\displaystyle = 7[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<tr id="a0000000259">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle g(3)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle = 4[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \qquad g'(3)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = 5[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
<tr id="a0000000260">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle g(5)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle = 3[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \qquad g'(5)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = 4[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
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<p>
If [mathjaxinline]h(x) = g(f(x))[/mathjaxinline], what is [mathjaxinline]h'(3)[/mathjaxinline]? </p>
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If [mathjaxinline]j(x) = f(g(x))[/mathjaxinline], what is [mathjaxinline]j'(3)[/mathjaxinline]? </p>
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Review 3
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In a circuit with resistance 10 ohms, power [mathjaxinline]P[/mathjaxinline] (measured in watts) is equal to [mathjaxinline]10 I^2[/mathjaxinline], where [mathjaxinline]I[/mathjaxinline] is the current measured in amperes. When [mathjaxinline]I = 6[/mathjaxinline] amperes, what is the rate of change of [mathjaxinline]P[/mathjaxinline] with respect to [mathjaxinline]I[/mathjaxinline]? </p>
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What units is your answer measured in? </p>
<p>
(Enter [mathjaxinline]o[/mathjaxinline] for ohms, [mathjaxinline]w[/mathjaxinline] for watts, [mathjaxinline]a[/mathjaxinline] for amperes, and [mathjaxinline]min[/mathjaxinline] for minutes. Type [mathjaxinline]*[/mathjaxinline] for multiplication; / for division; [mathjaxinline]\wedge[/mathjaxinline] for exponents.) </p>
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If we additionally know that when [mathjaxinline]I=6[/mathjaxinline], the current is increasing at 0.2 amperes per minute, at what rate (with respect to time) is power increasing at that moment? </p>
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What units is your answer measured in? </p>
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(Enter [mathjaxinline]o[/mathjaxinline] for ohms, [mathjaxinline]w[/mathjaxinline] for watts, [mathjaxinline]a[/mathjaxinline] for amperes, and [mathjaxinline]min[/mathjaxinline] for minutes. Type [mathjaxinline]*[/mathjaxinline] for multiplication; / for division; [mathjaxinline]\wedge[/mathjaxinline] for exponents.) </p>
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<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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<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
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<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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<h2 class="hd hd-2 unit-title">13. Practice the chain rule</h2>
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<h2 class="hd hd-2 unit-title">14. Summary</h2>
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<p><b class="bfseries">The Chain Rule</b></p><p>
If [mathjaxinline]h(x) = f(g(x))[/mathjaxinline], then </p><table id="a0000000261" 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'\left(g(x)\right) g'(x)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
at all points where the derivatives [mathjaxinline]\ f'(g(x))[/mathjaxinline] and [mathjaxinline]g'(x)[/mathjaxinline] are defined. </p><p>
Alternatively, if [mathjaxinline]y = f(u)[/mathjaxinline], and [mathjaxinline]u=g(x)[/mathjaxinline], then </p><table id="a0000000262" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\left. \frac{dy}{dx} \right|_{x = a} = \left. \frac{dy}{du} \right|_{u=g(a)} \left. \frac{du}{dx} \right|_{x=a}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
at any point [mathjaxinline]x=a[/mathjaxinline] where the derivatives on the right hand side are defined. </p>
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