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<h2 class="hd hd-2 unit-title">1. Motivation</h2>
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<h3 class="hd hd-2">Logarithms and their derivatives</h3>
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<h2 class="hd hd-2 unit-title">2. Logarithms and their derivatives</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 and their inverses<br/>products of basic functions<br/>quotients of basic functions <br/>compositions of basic functions <br/>implicit functions<br/>inverses of basic functions<br/>the exponential function </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 the <span style="color:#27408C"><b class="bf">logarithm function</b></span>. </p></li><li><p>
Identify the derivative of [mathjaxinline]a^ x[/mathjaxinline] explicitly. </p></li><li><p>
Use the <span style="color:#27408C"><b class="bf">inverse relationship</b></span> of [mathjaxinline]e[/mathjaxinline] and [mathjaxinline]\ln[/mathjaxinline] to differentiate complicated exponents. </p></li><li><p>
Use <span style="color:#27408C"><b class="bf">logarithmic differentiation</b></span> to deal with functions whose exponents are varying. </p></li></ul><p><b class="bfseries">Contents: 14 pages</b></p><p>
8 videos (42 minutes 1x speed) 16 questions </p>
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<h2 class="hd hd-2 unit-title">3. Preliminaries</h2>
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The exponential function
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Let [mathjaxinline]\ f(x) = e^ x[/mathjaxinline]. The derivative of [mathjaxinline]\ f[/mathjaxinline] is </p>
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Is [mathjaxinline]\ f[/mathjaxinline] one-to-one? </p>
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<h3 class="hd hd-2">Logarithms defined</h3>
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<p><b class="bfseries">Properties of [mathjaxinline]x[/mathjaxinline]</b></p><p>
[mathjaxinline]\log _{10}(x)[/mathjaxinline] is the inverse function of [mathjaxinline]10^ x[/mathjaxinline]. </p><p>
The natural log, denoted [mathjaxinline]\ln (x)[/mathjaxinline], is the inverse function of [mathjaxinline]e^ x[/mathjaxinline]. </p><ul class="itemize"><li><p>
[mathjaxinline]\ln e^ x =x[/mathjaxinline] </p></li><li><p>
[mathjaxinline]e^{\ln x} = x[/mathjaxinline] </p></li><li><p>
[mathjaxinline]\ln (ab) = \ln (a) + \ln (b)[/mathjaxinline] </p></li><li><p>
[mathjaxinline]\ln (a^ b) = b\ln (a)[/mathjaxinline] </p></li></ul>
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<h2 class="hd hd-2 unit-title">4. Properties of the logarithm function</h2>
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Log base 10
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What is [mathjaxinline]\log _{10} 1000[/mathjaxinline]? </p>
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What is [mathjaxinline]\log _{10} 0.000001[/mathjaxinline]? </p>
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Digits
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Suppose that [mathjaxinline]x[/mathjaxinline] is an integer with [mathjaxinline]\log _{10} x \approx 8.23[/mathjaxinline]. When written out, how many digits would [mathjaxinline]x[/mathjaxinline] have? </p>
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Estimate a natural logarithm
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Recall that [mathjaxinline]e \approx 2.71828.[/mathjaxinline] In fact, </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000378" style="table-layout:auto" width="100%">
<tr id="a0000000379">
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[mathjaxinline]\displaystyle e^2[/mathjaxinline]
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[mathjaxinline]\displaystyle \approx 7.39[/mathjaxinline]
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<tr id="a0000000380">
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[mathjaxinline]\displaystyle e^3[/mathjaxinline]
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[mathjaxinline]\displaystyle \approx 20.09[/mathjaxinline]
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</tr>
<tr id="a0000000381">
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[mathjaxinline]\displaystyle e^4[/mathjaxinline]
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[mathjaxinline]\displaystyle \approx 54.60[/mathjaxinline]
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[mathjaxinline]\displaystyle e^5[/mathjaxinline]
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[mathjaxinline]\displaystyle \approx 148.41[/mathjaxinline]
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[mathjaxinline]\displaystyle e^6[/mathjaxinline]
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[mathjaxinline]\displaystyle \approx 403.43[/mathjaxinline]
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[mathjaxinline]\displaystyle e^7[/mathjaxinline]
</td>
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[mathjaxinline]\displaystyle \approx 1096.63[/mathjaxinline]
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</tr>
</table>
<p>
Which of the following are between [mathjaxinline]5[/mathjaxinline] and [mathjaxinline]10[/mathjaxinline]? </p>
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(Click all possible answers.) </p>
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<text>[mathjaxinline]\ln 8[/mathjaxinline]</text>
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<text>[mathjaxinline]\ln 20[/mathjaxinline]</text>
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<text>[mathjaxinline]\ln 100[/mathjaxinline]</text>
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<text>[mathjaxinline]\ln 1000[/mathjaxinline]</text>
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Estimate a natural logarithm again
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Note that [mathjaxinline]10 \approx e^{2.3026}[/mathjaxinline]. A googol is the number [mathjaxinline]10^{100}[/mathjaxinline] (and yes, that is spelled correctly; incidentally, this number is greater than the number of atoms in the universe). What is the natural logarithm of one googol? </p>
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Estimate a natural logarithm one more time
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For what [mathjaxinline]x[/mathjaxinline] is [mathjaxinline]\ln x = 0[/mathjaxinline]? </p>
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<h2 class="hd hd-2 unit-title">5. Domain and Range</h2>
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Domain and range
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Recall the properties of exponential functions. </p>
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What is the domain of the function [mathjaxinline]e^ x[/mathjaxinline]? </p>
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<text> All positive numbers, plus zero</text>
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<text> All numbers that are not zero</text>
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What is the image (range) of the function [mathjaxinline]e^ x[/mathjaxinline]? </p>
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<text> All positive numbers</text>
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<text> All positive numbers, plus zero</text>
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Range and domain
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Almost all of the properties of [mathjaxinline]\ln x[/mathjaxinline] can be deduced from the corresponding properties of the exponential function [mathjaxinline]e^ x[/mathjaxinline]. Given the answers to the previous question, what is the domain of the function [mathjaxinline]\ln x[/mathjaxinline]? </p>
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<input type="radio" name="input_diff_8-tab5-problem2_2_1" id="input_diff_8-tab5-problem2_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="diff_8-tab5-problem2_2_1-choice_1-label" for="input_diff_8-tab5-problem2_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_diff_8-tab5-problem2_2_1">
<text> All real numbers</text>
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<text> All positive numbers, plus zero</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_8-tab5-problem2_2_1" id="input_diff_8-tab5-problem2_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="diff_8-tab5-problem2_2_1-choice_4-label" for="input_diff_8-tab5-problem2_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diff_8-tab5-problem2_2_1">
<text> All numbers that are not zero</text>
</label>
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<input type="radio" name="input_diff_8-tab5-problem2_2_1" id="input_diff_8-tab5-problem2_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="diff_8-tab5-problem2_2_1-choice_5-label" for="input_diff_8-tab5-problem2_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_diff_8-tab5-problem2_2_1">
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</p>
<p>
What is the image (range) of the function [mathjaxinline]\ln x[/mathjaxinline]? </p>
<p>
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<text> All real numbers</text>
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<text> All positive numbers</text>
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<text> All positive numbers, plus zero</text>
</label>
</div>
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<input type="radio" name="input_diff_8-tab5-problem2_3_1" id="input_diff_8-tab5-problem2_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="diff_8-tab5-problem2_3_1-choice_4-label" for="input_diff_8-tab5-problem2_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diff_8-tab5-problem2_3_1">
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Asymptotics
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<p>
We know that if [mathjaxinline]x[/mathjaxinline] gets really large and positive, then [mathjaxinline]e^ x[/mathjaxinline] will also get large and positive, without bound. When [mathjaxinline]x[/mathjaxinline] gets really large and negative (i.e., goes towards [mathjaxinline]-\infty[/mathjaxinline]), then [mathjaxinline]e^ x[/mathjaxinline]... </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_8-tab5-problem3_2_1">
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<text> gets really large and negative</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_8-tab5-problem3_2_1" id="input_diff_8-tab5-problem3_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diff_8-tab5-problem3_2_1-choice_2-label" for="input_diff_8-tab5-problem3_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diff_8-tab5-problem3_2_1">
<text> is negative, getting closer and closer to zero</text>
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<text> is positive, getting closer and closer to zero</text>
</label>
</div>
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<text> gets really large and positive</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diff_8-tab5-problem3_2_1" id="input_diff_8-tab5-problem3_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="diff_8-tab5-problem3_2_1-choice_5-label" for="input_diff_8-tab5-problem3_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_diff_8-tab5-problem3_2_1">
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<p>
<p style="display:inline">The corresponding fact about [mathjaxinline]\ln x[/mathjaxinline] is that when [mathjaxinline]x[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div class="inputtype option-input inline">
<select name="input_diff_8-tab5-problem3_3_1" id="input_diff_8-tab5-problem3_3_1" aria-describedby="status_diff_8-tab5-problem3_3_1">
<option value="option_diff_8-tab5-problem3_3_1_dummy_default">Select an option</option>
<option value="gets really large and negative"> gets really large and negative</option>
<option value="is negative, getting closer and closer to zero"> is negative, getting closer and closer to zero</option>
<option value="is positive, getting closer and closer to zero"> is positive, getting closer and closer to zero</option>
</select>
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<p class="answer" id="answer_diff_8-tab5-problem3_3_1"/>
</div></div>
<br/>
</p>
<p>
<p style="display:inline">then [mathjaxinline]\ln x[/mathjaxinline] </p>
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<select name="input_diff_8-tab5-problem3_4_1" id="input_diff_8-tab5-problem3_4_1" aria-describedby="status_diff_8-tab5-problem3_4_1">
<option value="option_diff_8-tab5-problem3_4_1_dummy_default">Select an option</option>
<option value="gets really large and negative"> gets really large and negative</option>
<option value="is negative, getting closer and closer to zero"> is negative, getting closer and closer to zero</option>
<option value="is positive, getting closer and closer to zero"> is positive, getting closer and closer to zero</option>
<option value="gets really large and positive"> gets really large and positive</option>
</select>
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<span class="status unanswered" id="status_diff_8-tab5-problem3_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
<p class="answer" id="answer_diff_8-tab5-problem3_4_1"/>
</div></div>
<br/>
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<span id="solution_diff_8-tab5-problem3_solution_1"/>
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<h2 class="hd hd-2 unit-title">6. Differentiating the logarithm</h2>
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<h2 class="hd hd-2 unit-title">7. The derivative of the natural logarithm</h2>
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Our collection of basic functions that we can differentiate now has one more entry: </p><table id="a0000000385" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\frac{d}{dx} \ln x = \frac{1}{x}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table>
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<h2 class="hd hd-2 unit-title">8. Differentiating exponential functions</h2>
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We found that [mathjaxinline]\displaystyle \frac{d}{dx} a^ x = M(a) a^ x[/mathjaxinline]. It is time to determine the mystery number [mathjaxinline]M(a)[/mathjaxinline]. </p><p>
To differentiate many exponential functions, the natural logarithm appears, well, naturally. </p><p>
There are two methods, which are essentially identical that can be used. We will learn these methods in the next two videos, and work through many many examples to help you get comfortable using these methods. </p>
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<h3 class="hd hd-2">Method 1: Changing to base e</h3>
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<p><b class="bfseries">Differentiating exponential functions with other bases</b></p><p>
We can finally identify our mystery number, and differentiate exponential functions with any base. </p><p>
For any positive constant [mathjaxinline]a[/mathjaxinline], </p><table id="a0000000390" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\frac{d}{dx} a^ x = a^ x \ln a.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table>
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Find the derivative of [mathjaxinline]f(t) = 10^ t[/mathjaxinline]. </p>
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(Type [mathjaxinline]*[/mathjaxinline] for multiplication; type [mathjaxinline]/[/mathjaxinline] for division; type [mathjaxinline]\wedge[/mathjaxinline] for exponents. The function [mathjaxinline]\ln[/mathjaxinline] is known by the grader, e.g. ln(t). Use parentheses to denote order of operations.) </p>
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<p style="display:inline">[mathjaxinline]f'(t)=[/mathjaxinline]</p>
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Let [mathjaxinline]g(t)=\log _{10} t[/mathjaxinline]. Find [mathjaxinline]g^{\prime }(t)[/mathjaxinline]. </p>
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(Type [mathjaxinline]*[/mathjaxinline] for multiplication; type [mathjaxinline]/[/mathjaxinline] for division; type [mathjaxinline]\wedge[/mathjaxinline] for exponents. The function [mathjaxinline]\ln[/mathjaxinline] is known by the grader, e.g. ln(t). Use parentheses to denote order of operations.) </p>
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<h2 class="hd hd-2 unit-title">9. Method 2: Logarithmic differentiation</h2>
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Let [mathjaxinline]f(x) = x^ x[/mathjaxinline]. To calculate [mathjaxinline]f'(x)[/mathjaxinline], which of the following rules apply? </p>
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<text> The formula [mathjaxinline]\displaystyle \frac{d}{dx}a^ x = a^ x \ln a[/mathjaxinline]</text>
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<text> The power rule</text>
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<text> The product rule</text>
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<h3 class="hd hd-2">Method 2: Logarithmic Differentiation</h3>
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True or false: If [mathjaxinline]f(x) = x^{x}[/mathjaxinline], then when [mathjaxinline]x &gt; 0[/mathjaxinline], [mathjaxinline]f'(x) &gt; f(x)[/mathjaxinline]. </p>
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<h2 class="hd hd-2 unit-title">10. Proof of the power rule for all real numbers</h2>
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<p><b class="bfseries">Differentiation methods</b></p><p><h3>Method 1: Change to base e</h3></p><p>
To calculate [mathjaxinline]\displaystyle \left( f(x)^{g(x)} \right)^{\prime },[/mathjaxinline] </p><ul class="itemize"><li><p>
first change base [mathjaxinline]\displaystyle f(x)^{g(x)} = e^{g(x)\ln f(x)}[/mathjaxinline], </p></li><li><p>
then differentiate using the chain rule, and the derivatives of the exponential and natural logarithm functions. </p></li></ul><p><h3>Method 2: Logarithmic differentiation</h3></p><p>
To calculate [mathjaxinline]\displaystyle u^{\prime },[/mathjaxinline] </p><ul class="itemize"><li><p>
first calculate [mathjaxinline]\displaystyle \left(\ln (u)\right)^{\prime }[/mathjaxinline], </p></li><li><p>
then use the formula </p><table id="a0000000398" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \frac{u^{\prime }}{u} = \left(\ln (u)\right)^{\prime }[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
and solve for [mathjaxinline]u'[/mathjaxinline]. </p></li></ul>
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<p><b class="bfseries">Complete proof of the power rule for all real powers [mathjaxinline]x^r[/mathjaxinline]</b></p><p>
We can use both methods for differentiating exponential functions here to give a proof of the power rule for any power [mathjaxinline]r[/mathjaxinline], rational or irrational. Let's see how. </p><p>
Note that you may want to try to work these proofs out on your own first as practice using the rules we learned above. </p><p><div class="hideshowbox"><h4 onclick="hideshow(this);" style="margin: 0px">Proof by Method 1<span class="icon-caret-down toggleimage"/></h4><div class="hideshowcontent">Method 1 is to change the base to [mathjaxinline]e[/mathjaxinline]. <table id="a0000000399" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000400"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle x^ r[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \left(e^{\ln x}\right)^ r = e^{r\ln x}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.51)</td></tr></table> Apply the chain rule using [mathjaxinline]u = r\ln x[/mathjaxinline]. <table id="a0000000401" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000402"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \frac{d}{dx}e^ u[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\quad \frac{de^ u}{du}\frac{du}{dx}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.52)</td></tr><tr id="a0000000403"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \quad e^{r\ln x} \frac{r}{x}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.53)</td></tr><tr id="a0000000404"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\quad x^ r\frac{r}{x} = rx^{r-1}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.54)</td></tr></table> This is what we expected to find! And this proof holds for [mathjaxinline]r[/mathjaxinline] any number, rational or irrational. </div><p class="hideshowbottom" onclick="hideshow(this);" style="margin: 0px"><a href="javascript: {return false;}">Show</a></p></div></p><p><div class="hideshowbox"><h4 onclick="hideshow(this);" style="margin: 0px">Proof by Method 2<span class="icon-caret-down toggleimage"/></h4><div class="hideshowcontent"><p>
Method 2 is to apply the natural logarithm and differentiate. </p><table id="a0000000405" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000406"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \ln x^ r[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = r\ln x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.55)</td></tr></table><p>
The derivative is [mathjaxinline]\displaystyle \frac{r}{x}[/mathjaxinline]. Now we need to use the formula: </p><table id="a0000000407" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000408"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \frac{u^{\prime }}{u}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \quad \left(\ln u\right)^{\prime }[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.56)</td></tr><tr id="a0000000409"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \frac{\left( x^ r \right)^{\prime }}{x^ r}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \quad \frac{r}{x}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.57)</td></tr><tr id="a0000000410"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \left( x^ r \right)^{\prime }[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \frac{rx^ r}{x} = \quad rx^{r-1}.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.58)</td></tr></table><p>
Thus we find [mathjaxinline]\displaystyle \frac{d}{dx} x^ r = rx^{r-1}[/mathjaxinline], which is what we expected! And this proof holds for [mathjaxinline]r[/mathjaxinline] any number, rational or irrational. </p></div><p class="hideshowbottom" onclick="hideshow(this);" style="margin: 0px"><a href="javascript: {return false;}">Show</a></p></div></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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<h2 class="hd hd-2 unit-title">11. Practice derivatives of logs and exponents</h2>
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<h2 class="hd hd-2 unit-title">12. Additional practice</h2>
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Derivative check
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(Enter e for [mathjaxinline]e[/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>
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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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Different bases
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(Enter e for [mathjaxinline]e[/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/>
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Suppose in January, [mathjaxinline]t=1[/mathjaxinline], a lake is covered by 2 square meters of algae. Every month, the area covered by algae doubles. (Assume that the growth occurs continuously.) Find the growth rate as a function of time with units square meters per month. Enter t for [mathjaxinline]t[/mathjaxinline] in months. </p>
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(Type [mathjaxinline]*[/mathjaxinline] for multiplication; type [mathjaxinline]/[/mathjaxinline] for division; type [mathjaxinline]\wedge[/mathjaxinline] for exponents. The function [mathjaxinline]\ln[/mathjaxinline] is known by the grader, e.g. ln(t). Use parentheses to denote order of operations.) </p>
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<td class="formulainput">dx, dy</td>
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<h2 class="hd hd-2 unit-title">13. The number e revisited</h2>
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<p><b class="bfseries">The number [mathjaxinline]e[/mathjaxinline]</b></p><p>
We can use the general approach of taking the natural logarithm for more than finding derivatives. Here we will use this method to evaluate a limit, and get a numerical approximation for the number [mathjaxinline]e[/mathjaxinline]! </p>
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<h2 class="hd hd-2 unit-title">14. Summary</h2>
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<p><b class="bfseries">Properties of [mathjaxinline]x[/mathjaxinline]</b></p><p>
[mathjaxinline]\log _{10}(x)[/mathjaxinline] is the inverse function of [mathjaxinline]10^ x[/mathjaxinline]. </p><p>
The natural log, denoted [mathjaxinline]\ln (x)[/mathjaxinline], is the inverse function of [mathjaxinline]e^ x[/mathjaxinline]. </p><ul class="itemize"><li><p>
[mathjaxinline]\ln e^ x =x[/mathjaxinline] </p></li><li><p>
[mathjaxinline]e^{\ln x} = x[/mathjaxinline] </p></li><li><p>
[mathjaxinline]\ln (ab) = \ln (a) + \ln (b)[/mathjaxinline] </p></li><li><p>
[mathjaxinline]\ln (a^ b) = b\ln (a)[/mathjaxinline] </p></li></ul><p><b class="bfseries">The derivative of the natural logarithm</b></p><p>
Our collection of basic functions that we can differentiate now has one more entry: </p><table id="a0000000427" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\frac{d}{dx} \ln x = \frac{1}{x}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p><b class="bfseries">Differentiating exponential functions with other bases</b></p><p>
For any positive constant [mathjaxinline]a[/mathjaxinline], </p><table id="a0000000428" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\frac{d}{dx} a^ x = a^ x \ln a.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p><b class="bfseries">Differentiation methods</b></p><p><h3>Method 1: Change to base e</h3></p><p>
To calculate [mathjaxinline]\displaystyle \left( f(x)^{g(x)} \right)^{\prime },[/mathjaxinline] </p><ul class="itemize"><li><p>
first change base [mathjaxinline]\displaystyle f(x)^{g(x)} = e^{g(x)\ln f(x)}[/mathjaxinline], </p></li><li><p>
then differentiate using the chain rule, and the derivatives of the exponential and natural logarithm functions. </p></li></ul><p><h3>Method 2: Logarithmic differentiation</h3></p><p>
To calculate [mathjaxinline]\displaystyle u^{\prime },[/mathjaxinline] </p><ul class="itemize"><li><p>
first calculate [mathjaxinline]\displaystyle \left(\ln (u)\right)^{\prime }[/mathjaxinline], </p></li><li><p>
then use the formula </p><table id="a0000000429" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \frac{u^{\prime }}{u} = \left(\ln (u)\right)^{\prime }[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
and solve for [mathjaxinline]u'[/mathjaxinline]. </p></li></ul>
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