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<h2 class="hd hd-2 unit-title">1. Motivation</h2>
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<h3 class="hd hd-2">Implicit Differentiation</h3>
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<h2 class="hd hd-2 unit-title">2. Implicit Differentiation</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/>trigonometric functions <br/>products of basic functions<br/>quotients of basic functions <br/>compositions 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>
Recognize <span style="color:#27408C"><b class="bf">implicit functions</b></span>. </p></li><li><p><span style="color:#27408C"><b class="bf">Apply the chain rule</b></span> to differentiate implicit functions. </p></li><li><p>
Understand why the <span style="color:#27408C"><b class="bf">power rule for fractional powers</b></span> holds: </p><table id="a0000000263" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle {\frac{d}{dx}x^{m/n} = (m/n)x^{m/n-1}}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table></li></ul><p><b class="bfseries">Contents: 10 pages</b></p><p>
5 videos (23 minutes 1x speed) 14 questions </p>
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<h2 class="hd hd-2 unit-title">3. Differentiating the circle</h2>
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The equation [mathjaxinline]x^2 + y^2 = 25[/mathjaxinline] describes a circle of radius 5. </p><center><img src="/assets/courseware/v1/82d3eef6d79a6d1893803733e21b66a4/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_implicitcircle.svg" width="350px" alt="See text above." style="margin: 10px 25px 25px 25px"/></center><p>
This curve does not satisfy the vertical line test–there are two [mathjaxinline]y[/mathjaxinline] values for almost every [mathjaxinline]x[/mathjaxinline] value. But you can still solve for [mathjaxinline]y[/mathjaxinline]. Note that you won't be able to find one equation. You'll find two—one for the upper semicircle, and one for the lower semicircle. </p>
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Solve [mathjaxinline]x^2 + y^2 = 25[/mathjaxinline] to find an explicit formula for [mathjaxinline]y[/mathjaxinline] in terms of [mathjaxinline]x[/mathjaxinline]. You will have two expressions! Put the positive expression in the first box, and the negative expression in the second box. </p>
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(Enter sqrt for [mathjaxinline]\sqrt { }[/mathjaxinline]. Enter [mathjaxinline]\wedge[/mathjaxinline] to denote exponents, e.g. [mathjaxinline]x\wedge 2 = x^2[/mathjaxinline]. Use parentheses to denote order of operations.) </p>
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<p style="display:inline">Top half: &#8195;&#8195;&#8195;</p>
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<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
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<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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<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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<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">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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<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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Differentiate
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For each equation above, find [mathjaxinline]\displaystyle \frac{dy}{dx}[/mathjaxinline] and enter in the corresponding box below. </p>
<p>
(Enter sqrt for [mathjaxinline]\sqrt { }[/mathjaxinline]. Enter [mathjaxinline]\wedge[/mathjaxinline] to denote exponents, e.g. [mathjaxinline]x\wedge 2 = x^2[/mathjaxinline]. Use parentheses to denote order of operations. Enter the derivative of the top function in the top answer box; the derivative of the bottom function in the lower answer box.) </p>
<p>
<p style="display:inline">Top half derivative: &#8195;&#8195;&#8195;</p>
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<p style="display:inline">Bottom half derivative: &#8195;</p>
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<h2 class="hd hd-2 unit-title">4. Implicitly differentiate the circle</h2>
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Find the tangent line slope
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What is the slope of the tangent line to the circle [mathjaxinline]x^2 + y^2 =25[/mathjaxinline] at the point [mathjaxinline](-3,4)[/mathjaxinline]? </p>
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<p><b class="bfseries"><span style="color:#FF7F00">Note on video:</span></b> In the video above, we assume that [mathjaxinline]n \neq 0[/mathjaxinline] so that the quotient is defined, and [mathjaxinline]n\neq 1[/mathjaxinline] so that the quotient is nontrivial. </p>
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<h2 class="hd hd-2 unit-title">6. Implicit functions</h2>
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The idea behind the previous problem was that immediately differentiating [mathjaxinline]y=x^{m/n}[/mathjaxinline] was hard. But differentiating the <em>implicit</em> function [mathjaxinline]y^ n=x^ m[/mathjaxinline] was a whole lot easier. This happens any time your function is more simply described implicitly. An implicit function is an equation involving both [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]y[/mathjaxinline] (or any two variables really). You <em>could</em> solve for [mathjaxinline]y[/mathjaxinline] as a function of [mathjaxinline]x[/mathjaxinline], but often times, that computation is messy (or impossible). If a function is not implicit, we say that it is explicit. </p><p><b class="bfseries">Examples</b></p><ul class="itemize"><li><p>
[mathjaxinline]y^3=x^2[/mathjaxinline] is implicit. </p></li><li><p>
[mathjaxinline]y=x^{2/3}[/mathjaxinline] is explicit. </p></li><li><p>
[mathjaxinline]y^2=x[/mathjaxinline] is an implicit description of the two functions [mathjaxinline]y= \pm \sqrt {x}[/mathjaxinline]. </p></li></ul>
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Implicit or Explicit
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Which of the following equations give [mathjaxinline]y[/mathjaxinline] as an implicit function of [mathjaxinline]x[/mathjaxinline]? (Check as many as apply.) </p>
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<h2 class="hd hd-2 unit-title">7. Worked example: An equipotential curve</h2>
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The implicit function that describes this curve is [mathjaxinline]x^4 -3x^2 + y^4 + y^2 + 2x^2 y^2 = 0[/mathjaxinline]. </p><center><img src="/assets/courseware/v1/992cf3a8b16af9dba0f500b33d972afb/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_lemniscate.svg" width="350px" alt="See text above." style="margin: 10px 25px 25px 25px"/></center><p><div class="hideshowbox"><h4 onclick="hideshow(this);" style="margin: 0px">Context for this curve<span class="icon-caret-down toggleimage"/></h4><div class="hideshowcontent"><p>
Suppose we have two point charges at [mathjaxinline](\pm 1, 0)[/mathjaxinline]. The points where the potential energy of an electron would be equal to the potential energy at the origin is a curve called a lemniscate. </p></div><p class="hideshowbottom" onclick="hideshow(this);" style="margin: 0px"><a href="javascript: {return false;}">Show</a></p></div></p><p>
We can ask a lot of questions about this curve. For example: </p><ol class="enumerate"><li value="1"><p>
What is the slope of the tangent line at the point [mathjaxinline]\displaystyle \left(\frac1{\sqrt {2}},\frac1{\sqrt {2}}\right)[/mathjaxinline]?, or </p></li><li value="2"><p>
What are the points on this curve where the tangent line is horizontal? </p></li></ol><p>
Answering both of these questions requires finding [mathjaxinline]\frac{dy}{dx}[/mathjaxinline]. Rather than solve for [mathjaxinline]y[/mathjaxinline], we'll use implicit differentiation. Here are questions to help you get started in answering these questions. </p><SCRIPT src="/assets/courseware/v1/631e447105fca1b243137b21b9ed6f90/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/latex2edx.js" type="text/javascript"/><LINK href="/assets/courseware/v1/daf81af0af57b85a105e0ed27b7873a0/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/latex2edx.css" rel="stylesheet" type="text/css"/>
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Differentiating the first part
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We want to differentiate both sides of [mathjaxinline]x^4 -3x^2 + y^4 + y^2 + 2x^2 y^2 = 0[/mathjaxinline] with respect to [mathjaxinline]x[/mathjaxinline]. The last term on the left hand side will get tricky, but the first four aren't so bad. </p><p>
True or false: </p>[mathjax]\begin{equation}
\frac{d}{dx}\left(x^4 - 3x^2 + y^4 + y^2\right) = 4x^3 - 6x + 4y^3 + 2y.
\end{equation}[/mathjax]<p><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diff_5-tab7-problem1_2_1">
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Differentiating the last term
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When trying to differentiate [mathjaxinline]2x^2y^2[/mathjaxinline] with respect to [mathjaxinline]x[/mathjaxinline], which of the following do we need? Check as many as apply. </p>
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<text>Chain Rule</text>
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<h2 class="hd hd-2 unit-title">8. Exercises</h2>
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Find the errors
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Consider the curve defined by [mathjaxinline]2y + \cos x + x^2y + \sin 2y = 3[/mathjaxinline]. I tried to use implicit differentiation (with respect to [mathjaxinline]x[/mathjaxinline]) to find the slope of a tangent line. Below is the equation I got. Which of the terms in this equation were derived incorrectly? Drag the red <b class="bf">X</b> on top of every term that is incorrect. </p>
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Differentiate with respect to any variable
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Consider the curve defined by [mathjaxinline]y^3+x^3 = 3xy[/mathjaxinline]. </p>
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<img alt="See text above." src="/assets/courseware/v1/0a6e2a38b9e4dd320e5b078336ef5aa7/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_implicitloop.svg" style="margin: 10px 25px 25px 25px" width="150px"/>
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Think of [mathjaxinline]y=y(x)[/mathjaxinline] and differentiate to find [mathjaxinline]\frac{dy}{dx}[/mathjaxinline]. </p>
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(Type [mathjaxinline]*[/mathjaxinline] for multiplication; type [mathjaxinline]/[/mathjaxinline] for division; type [mathjaxinline]\wedge[/mathjaxinline] for exponents. Use parentheses to denote order of operations.) </p>
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<p style="display:inline">[mathjaxinline]\displaystyle \frac{dy}{dx}=[/mathjaxinline]</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 \(\lambda \)
</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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Find the derivative of [mathjaxinline]y^2 = x[/mathjaxinline] as a function of [mathjaxinline]y[/mathjaxinline]. </p>
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(Type [mathjaxinline]*[/mathjaxinline] for multiplication; type [mathjaxinline]/[/mathjaxinline] for division; type [mathjaxinline]\wedge[/mathjaxinline] for exponents. Use parentheses to denote order of operations.) </p>
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<p style="display:inline">[mathjaxinline]\displaystyle {\frac{dy}{dx}=}[/mathjaxinline]</p>
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Now find the derivative as a function of [mathjaxinline]x[/mathjaxinline]. Enter the positive function first. </p>
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(Type sqrt(x) for [mathjaxinline]\sqrt {x}[/mathjaxinline]. Type [mathjaxinline]*[/mathjaxinline] for multiplication; type [mathjaxinline]/[/mathjaxinline] for division; type [mathjaxinline]\wedge[/mathjaxinline] for exponents. Use parentheses to denote order of operations.) </p>
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<p style="display:inline">When [mathjaxinline]y&gt;0[/mathjaxinline]: &#8195;&#8195;[mathjaxinline]\displaystyle {\frac{dy}{dx}=}[/mathjaxinline]</p>
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<p style="display:inline">When [mathjaxinline]y&lt;0[/mathjaxinline]: &#8195;&#8195;[mathjaxinline]\displaystyle {\frac{dy}{dx}=}[/mathjaxinline]</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 \(\lambda \)
</td>
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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 \( 2\pi \)
</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 \( \sqrt{x^2-y} \)
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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>
</tr>
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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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Practice 2
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Find the derivative [mathjaxinline]\displaystyle {\frac{dy}{dx}}[/mathjaxinline] of the function [mathjaxinline]x^2 + 4xy =2y^2+5[/mathjaxinline]. </p>
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(Type [mathjaxinline]*[/mathjaxinline] for multiplication; type [mathjaxinline]/[/mathjaxinline] for division; type [mathjaxinline]\wedge[/mathjaxinline] for exponents. Use parentheses to denote order of operations.) </p>
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<p style="display:inline">[mathjaxinline]\displaystyle {\frac{dy}{dx}=}[/mathjaxinline]</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>
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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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<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>
</tr>
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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>
</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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Practice 3
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Find the derivative [mathjaxinline]\displaystyle {\frac{du}{dy}}[/mathjaxinline] of the function [mathjaxinline]u = \sin (y^2 + u)[/mathjaxinline]. </p>
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(Type sin(y&#8743;2 + u) and cos(y&#8743;2 + u) for the trig functions. Type [mathjaxinline]*[/mathjaxinline] for multiplication; type [mathjaxinline]/[/mathjaxinline] for division; type [mathjaxinline]\wedge[/mathjaxinline] for exponents. Use parentheses to denote order of operations.) </p>
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<p style="display:inline">[mathjaxinline]\displaystyle {\frac{du}{dy}=}[/mathjaxinline]</p>
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<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>
</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>
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<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>
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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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Practice 4
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Find the derivative [mathjaxinline]\displaystyle {\frac{dw}{dv}}[/mathjaxinline] of the function [mathjaxinline]w^2v^3 = w^3v^2[/mathjaxinline]. </p>
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(Type [mathjaxinline]*[/mathjaxinline] for multiplication; type [mathjaxinline]/[/mathjaxinline] for division; type [mathjaxinline]\wedge[/mathjaxinline] for exponents. Use parentheses to denote order of operations.) </p>
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<p style="display:inline">[mathjaxinline]\displaystyle {\frac{dw}{dv}=}[/mathjaxinline]</p>
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
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<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>
</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>
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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 \( 2\pi \)
</td>
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<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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<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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Practice 5
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Find the derivative [mathjaxinline]\displaystyle {\frac{dy}{dx}}[/mathjaxinline] of the function [mathjaxinline]xy = y^3[/mathjaxinline]. </p>
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(Type [mathjaxinline]*[/mathjaxinline] for multiplication; type [mathjaxinline]/[/mathjaxinline] for division; type [mathjaxinline]\wedge[/mathjaxinline] for exponents. Use parentheses to denote order of operations.) </p>
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<p style="display:inline">[mathjaxinline]\displaystyle {\frac{dy}{dx}=}[/mathjaxinline]</p>
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<th class="formulainput" scope="col">Descriptions</th>
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<font color="#0078b0">2520</font>
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<font color="#0078b0">2/3</font>
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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>
</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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Circle Revisited
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On the circle [mathjaxinline]x^2 + y^2 = 25[/mathjaxinline], we saw that [mathjaxinline]\displaystyle {\frac{dy}{dx} = -\frac{x}{y}}[/mathjaxinline]. What is [mathjaxinline]\displaystyle {\frac{d^2y}{dx^2}}[/mathjaxinline] at the point [mathjaxinline](-3,4)[/mathjaxinline]? </p>
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(Enter as a decimal or fraction.) </p>
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<th class="formulainput" scope="col">Allowable Entries</th>
<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>
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<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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<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>
<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>
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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">11. Summary</h2>
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<p><b class="bfseries">Implicit functions</b></p><p>
Immediately differentiating [mathjaxinline]y=x^{m/n}[/mathjaxinline] is hard. But differentiating the <em>implicit</em> function [mathjaxinline]y^ n=x^ m[/mathjaxinline] was a whole lot easier. This happens any time your function is more simply described implicitly. </p><p>
An <span style="color:#27408C"><b class="bf">implicit function</b></span> is an equation involving both [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]y[/mathjaxinline] (or any two variables really); you <em>could</em> solve for [mathjaxinline]y[/mathjaxinline] as a function of [mathjaxinline]x[/mathjaxinline], but often times, that computation is messy (or impossible). If a function is not implicit, we say that it is explicit. </p>
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