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<h2 class="hd hd-2 unit-title">1. Motivation</h2>
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<h3 class="hd hd-2">The Quotient Rule</h3>
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<h2 class="hd hd-2 unit-title">2. The quotient rule</h2>
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<p><b class="bfseries">What functions can we differentiate so far?</b></p><p>
powers of [mathjaxinline]x[/mathjaxinline]<br/>polynomials<br/>sine and cosine <br/>products 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:#27408C"><b class="bf">quotients of functions</b></span>. </p></li><li><p>
Find derivatives of all <span style="color:#27408C"><b class="bf">trigonometric functions</b></span>, including secant and tangent. </p></li><li><p>
Derive the power rule for negative integer powers. </p></li></ul><p><b class="bfseries">Contents: 11 pages</b></p><p>
5 videos (12 minutes 1x speed) 13 questions </p>
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<h2 class="hd hd-2 unit-title">3. Setting the stage</h2>
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Let [mathjaxinline]\ f(t)[/mathjaxinline] and [mathjaxinline]g(t)[/mathjaxinline] be functions, and let [mathjaxinline]h(t) = f(t)\, g(t)[/mathjaxinline]. </p>
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Is [mathjaxinline]h'(t)[/mathjaxinline] equal to the product of [mathjaxinline]\ f'(t)[/mathjaxinline] and [mathjaxinline]g'(t)[/mathjaxinline]? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diff_3-tab3-problem1_2_1">
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Now let [mathjaxinline]h(t) = f(t)/g(t)[/mathjaxinline]. Do you expect that [mathjaxinline]h'(t)[/mathjaxinline] will be the quotient [mathjaxinline]\ f'(t)/g'(t)[/mathjaxinline]? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diff_3-tab3-problem1_3_1">
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Say [mathjaxinline]t[/mathjaxinline] is measured in seconds, [mathjaxinline]\ f(t)[/mathjaxinline] in volts, and [mathjaxinline]g(t)[/mathjaxinline] in meters. What are the units of [mathjaxinline]\displaystyle h(t) = \frac{f(t)}{g(t)}[/mathjaxinline]? </p>
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(Type [mathjaxinline]s[/mathjaxinline] for seconds, [mathjaxinline]V[/mathjaxinline] for volts, and [mathjaxinline]m[/mathjaxinline] for meters. Type [mathjaxinline]*[/mathjaxinline] for multiplication; type [mathjaxinline]/[/mathjaxinline] for division. Use parentheses to denote order of operations.) </p>
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What are the units of [mathjaxinline]\displaystyle \frac{dh}{dt}[/mathjaxinline]? Use [mathjaxinline]s[/mathjaxinline] for seconds, [mathjaxinline]V[/mathjaxinline] for volts, and [mathjaxinline]m[/mathjaxinline] for meters. </p>
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(Type [mathjaxinline]*[/mathjaxinline] for multiplication; type [mathjaxinline]/[/mathjaxinline] for division. Use parentheses to denote order of operations.) </p>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_diff_3-tab3-problem2_3_1" class="inputtype formulaequationinput">
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What are the units of the quantity [mathjaxinline]\displaystyle \frac{\frac{df}{dt}}{\frac{dg}{dt}}[/mathjaxinline]? </p>
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(Type [mathjaxinline]*[/mathjaxinline] for multiplication; type [mathjaxinline]/[/mathjaxinline] for division. Use parentheses to denote order of operations.) </p>
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<font color="#0078b0">2520</font>
</td>
</tr>
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<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</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>
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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>
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<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
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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 \(\lambda \)
</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>
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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</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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Identify the change in h
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<p>
Let [mathjaxinline]h(t) = f(t)/g(t)[/mathjaxinline]. In terms of [mathjaxinline]\ f, g, \Delta f,[/mathjaxinline] and [mathjaxinline]\Delta g[/mathjaxinline], what is [mathjaxinline]\Delta h[/mathjaxinline]? </p>
<p>
(If you don't remember how [mathjaxinline]\Delta [/mathjaxinline] notation is used, you may wish to revisit the <a href="/courses/course-v1:MITx+18.01.1x+2T2019/jump_to_id/diff_2-tab10">product rule video</a>.)</p>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diff_3-tab4-problem1_2_1">
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<text> [mathjaxinline]\displaystyle \frac{\Delta f}{\Delta g}[/mathjaxinline] <br/></text>
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<text> [mathjaxinline]\displaystyle \frac{\Delta f}{g}[/mathjaxinline] <br/></text>
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<text> [mathjaxinline]\displaystyle \frac{f}{g} + \frac{\Delta f}{\Delta g}[/mathjaxinline] <br/></text>
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<text> [mathjaxinline]\displaystyle \frac{f+\Delta f}{g}[/mathjaxinline] <br/></text>
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<text> [mathjaxinline]\displaystyle \frac{f + \Delta f}{g + \Delta g}[/mathjaxinline] <br/></text>
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<text> [mathjaxinline]\displaystyle \frac{f + \Delta f}{g + \Delta g} - \frac{f}{g}[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">5. Next steps</h2>
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Identifying the Limit 1
</h3>
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<p>
We'll find a common denominator to simplify the expression you found into a single term: </p>
<table cellpadding="7" cellspacing="0" class="equation" id="a0000000216" style="table-layout:auto" width="100%">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\Delta h = \frac{(f+\Delta f)g}{(g + \Delta g)g}- \frac{f(g+\Delta g)}{(g + \Delta g)g}= \frac{\Delta f \cdot g - f \cdot \Delta g}{ g^2 + g \cdot \Delta g}.[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
Of course we still need to divide by [mathjaxinline]\Delta t[/mathjaxinline] and then take a limit as [mathjaxinline]\Delta t[/mathjaxinline] goes to zero. Here's what we get when we divide by [mathjaxinline]\Delta t[/mathjaxinline]: </p>
<table cellpadding="7" cellspacing="0" class="equation" id="a0000000217" style="table-layout:auto" width="100%">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\frac{\Delta h}{\Delta t} =\frac{(g \cdot \Delta f - f \cdot \Delta g)}{\Delta t(g^2 + g \cdot \Delta g)}= \frac{\frac{(g \cdot \Delta f - f \cdot \Delta g)}{\Delta t}}{g^2 + g \cdot \Delta g}.[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
What is the limit of [mathjaxinline]\displaystyle \frac{\Delta f \cdot g - f \cdot \Delta g}{\Delta t}[/mathjaxinline] as [mathjaxinline]\Delta t \rightarrow 0[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diff_3-tab5-problem1_2_1">
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<text> [mathjaxinline]\displaystyle g' \cdot f'[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle g' - f'[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle f\cdot g - f' \cdot g'[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle g\cdot f' - f' \cdot g[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle f\cdot g' - g \cdot f'[/mathjaxinline]</text>
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What is the limit of [mathjaxinline]g^2 + g \cdot \Delta g[/mathjaxinline] as [mathjaxinline]\Delta t \rightarrow 0[/mathjaxinline]? </p>
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<text> [mathjaxinline]\displaystyle g^2 + g'[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle g^2 + g \cdot g'[/mathjaxinline]</text>
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<h3 class="hd hd-2">The Quotient Rule and Worked Example</h3>
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<h2 class="hd hd-2 unit-title">6. The Quotient Rule</h2>
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If [mathjaxinline]\displaystyle h(x) = \frac{f(x)}{g(x)}[/mathjaxinline] for all [mathjaxinline]x[/mathjaxinline], then </p><table id="a0000000221" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle h'(x) = \frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
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Quotient rule practice
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<h2 class="hd hd-2 unit-title">7. Making sense of the quotient rule</h2>
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The numerator of the quotient rule looks like the product rule, except one of the terms is <span style="color:#27408C"><p>
subtracted from the other. This makes it important to remember which term comes first! The next few questions will help you develop an intuition about which term is subtracted and why. </p></span></p>
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Changing the numerator
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<p>
Assume that both [mathjaxinline]\ f(x)[/mathjaxinline] and [mathjaxinline]g(x)[/mathjaxinline] are positive. </p>
<p>
If [mathjaxinline]\ f(x)[/mathjaxinline] is increasing, holding [mathjaxinline]g[/mathjaxinline] fixed, would that increase the quotient [mathjaxinline]h(x) = f(x)/g(x)[/mathjaxinline] or decrease it? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diff_3-tab7-problem1_2_1">
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<text> Increase</text>
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<text> Decrease</text>
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<text> Neither</text>
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<p>
Based on your answer, and still holding [mathjaxinline]g[/mathjaxinline] fixed, should the sign of [mathjaxinline]\ f'(x)[/mathjaxinline] be the same as the sign of [mathjaxinline]h'(x)[/mathjaxinline], or opposite? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diff_3-tab7-problem1_3_1">
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<text> Same sign</text>
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<text> Opposite sign</text>
</label>
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<text> Neither</text>
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Changing the denominator
</h3>
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<div class="problem">
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<p>
Again assume that both [mathjaxinline]\ f(x)[/mathjaxinline] and [mathjaxinline]g(x)[/mathjaxinline] are positive. </p>
<p>
If [mathjaxinline]g(x)[/mathjaxinline] is increasing, holding [mathjaxinline]\ f[/mathjaxinline] fixed, would that increase the quotient [mathjaxinline]h(x) = \frac{f(x)}{g(x)}[/mathjaxinline] or decrease it? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diff_3-tab7-problem2_2_1">
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<text> Increase</text>
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<text> Decrease</text>
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<text> Neither</text>
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</p>
<p>
Based on your answer, and still holding [mathjaxinline]\ f[/mathjaxinline] fixed, should the sign of [mathjaxinline]g'(x)[/mathjaxinline] be the same as the sign of [mathjaxinline]h'(x)[/mathjaxinline], or opposite? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diff_3-tab7-problem2_3_1">
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<text> Same sign</text>
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<text> Opposite sign</text>
</label>
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<text> Neither</text>
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<p><b class="bfseries">Remembering the quotient rule</b></p><p>
You may want to find a mnemonic for remembering the quotient rule. </p><p>
One way to remember it is to put the following rhyme to song: </p><center> The derivative of the top times the bottom, <br/>minus the top times the derivative of the bottom, <br/>now don't you be scared, <br/>but over the bottom it is squared. <br/>This is calculus; <br/>this is calculus; <br/>this is calculus. </center><p>
Share your own techniques for remembering the quotient rule! </p>
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<h2 class="hd hd-2 unit-title">8. Power rule revisited</h2>
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Concept check
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If [mathjaxinline]\ f(x)[/mathjaxinline] is non-zero and differentiable, then the derivative of the function [mathjaxinline]1/f(x)[/mathjaxinline] is [mathjaxinline]1/f'(x)[/mathjaxinline]. <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diff_3-tab8-problem1_2_1">
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Quotient rule practice 2
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Recall that [mathjaxinline]\displaystyle \sec x = \frac{1}{\cos x}[/mathjaxinline]. Find the derivative of [mathjaxinline]\sec x[/mathjaxinline]. </p>
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(The functions sin, cos, tan, sec, csc, cot are all known by the grader; e.g. <b class="bf">sin(x)</b>. Enter the square of a trig function [mathjaxinline]\sin ^2(x)[/mathjaxinline] as <b class="bf">(sin(x))^2</b>.) </p>
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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/>
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<h2 class="hd hd-2 unit-title">9. Practice the quotient rule</h2>
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<table id="a0000000225" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000226"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle {\frac{d}{dx}} \sin x[/mathjaxinline]
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[mathjaxinline]\displaystyle = \cos x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000227"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle {\frac{d}{dx}} \cos x[/mathjaxinline]
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[mathjaxinline]\displaystyle = -\sin x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000228"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle {\frac{d}{dx}} \tan x[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \frac{1}{\cos ^2 x} = \sec ^2 x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000229"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle {\frac{d}{dx}} \cot x[/mathjaxinline]
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[mathjaxinline]\displaystyle = -\frac{1}{\sin ^2 x} = -\csc ^2 x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000230"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle {\frac{d}{dx}} \sec x[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \frac{\sin x}{\cos ^2 x} = \sec x \tan x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000231"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle {\frac{d}{dx}} \csc x[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = -\frac{\cos x}{\sin ^2 x} = -\csc x \cot x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table>
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<h2 class="hd hd-2 unit-title">11. Fundamentals</h2>
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What rule should we use?
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What is the most convenient rule to use to differentiate [mathjaxinline]\displaystyle \frac{5}{x^4}[/mathjaxinline]? </p>
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<text> Power Rule</text>
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<text> Quotient Rule</text>
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What is the most convenient rule to use to differentiate [mathjaxinline]\displaystyle \frac{x^2}{x^3+1}[/mathjaxinline]? </p>
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What is the most convenient rule to use to differentiate [mathjaxinline]\displaystyle \frac{x+\sqrt {x}}{\cos (1)}[/mathjaxinline]? </p>
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One more function
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Can we use the quotient rule to differentiate [mathjaxinline]\sin (1/x)[/mathjaxinline]? </p>
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<h2 class="hd hd-2 unit-title">12. Summary</h2>
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<p><b class="bfseries">The quotient rule</b></p><p>
If [mathjaxinline]\displaystyle h(x) = \frac{f(x)}{g(x)}[/mathjaxinline] for all [mathjaxinline]x[/mathjaxinline], then </p><table id="a0000000232" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle h'(x) = \frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
at all points where [mathjaxinline]\ f[/mathjaxinline] and [mathjaxinline]g[/mathjaxinline] are differentiable and [mathjaxinline]g(x) \ne 0[/mathjaxinline]. </p><p><b class="bfseries">Derivatives of all trigonometric functions</b></p><table id="a0000000233" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000234"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle {\frac{d}{dx}} \sin x[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \cos x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000235"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle {\frac{d}{dx}} \cos x[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = -\sin x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000236"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle {\frac{d}{dx}} \tan x[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \frac{1}{\cos ^2 x} = \sec ^2 x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000237"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle {\frac{d}{dx}} \cot x[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = -\frac{1}{\sin ^2 x} = -\csc ^2 x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000238"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle {\frac{d}{dx}} \sec x[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \frac{\sin x}{\cos ^2 x} = \sec x \tan x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000239"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle {\frac{d}{dx}} \csc x[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = -\frac{\cos x}{\sin ^2 x} = -\csc x \cot x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table>
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