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<h2 class="hd hd-2 unit-title">1. Leibniz notation</h2>
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<h3 class="hd hd-2">Leibniz Notation</h3>
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<p><b class="bfseries"><span style="color:#FF7F00">Note on video:</span></b> Technically, Newton developed dot notation ([mathjaxinline]\dot f(t)[/mathjaxinline]), which is only used with derivatives with respect to time, and Lagrange developed prime notation ([mathjaxinline]f'(x)[/mathjaxinline]). </p>
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<h2 class="hd hd-2 unit-title">2. Objectives</h2>
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<table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:center; border:none">
[mathjaxinline]f'(x)[/mathjaxinline] </td><td style="text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{df}{dx}[/mathjaxinline] </td><td style="text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{dy}{dx}[/mathjaxinline] </td><td style="text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{d}{dx}f[/mathjaxinline] </td><td style="text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{d}{dx}y[/mathjaxinline] </td></tr></table><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>
Understand the rationales for <span style="color:#99182C"><b class="bf">Leibniz</b></span> notation. </p></li><li><p>
Draw <span style="color:#99182C"><b class="bf">differential elements</b></span>. </p></li><li><p>
Be able to use Leibniz notation appropriately. </p></li></ul><p><b class="bfseries">Contents: 7 pages</b></p><p>
3 videos (7 minutes 1x speed) 10 questions </p>
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<h2 class="hd hd-2 unit-title">3. Notation</h2>
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<h3 class="hd hd-2">Introducing Leibniz Notation</h3>
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<td class="equation" style="width:80%; border:none">[mathjax]f'(x) = \lim _{\Delta x\rightarrow 0} \frac{\Delta f}{\Delta x}[/mathjax]</td>
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What does the letter [mathjaxinline]d[/mathjaxinline] in Leibniz's notation [mathjaxinline]\displaystyle \frac{df}{dx}[/mathjaxinline] stand for? </p>
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<text> [mathjaxinline]d[/mathjaxinline] stands for derivative</text>
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<text> [mathjaxinline]d[/mathjaxinline] stands for difference</text>
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<text> [mathjaxinline]d[/mathjaxinline] stands for the limit of a difference</text>
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<text> [mathjaxinline]d[/mathjaxinline] stands for dragon</text>
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<h2 class="hd hd-2 unit-title">4. Derivatives of areas of circles</h2>
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Why do we like to use Leibniz notation? </p><p>
The biggest reason is that it reminds us what the input variable is. The derivative is measuring the instantaneous rate of change of the output variable of a function with respect to the input variable. Sometimes, if there are lots of quantities that have variables representing them, it's easy to lose track of what is what; Leibniz notation helps to remind us. </p><p>
The following set of questions will help you understand why this is useful. </p>
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Area problem 1
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Write down the formula for the area [mathjaxinline]A[/mathjaxinline] of a circle in terms of the radius [mathjaxinline]r[/mathjaxinline]. Then take the derivative, and evaluate it at a point. </p>
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(Type <b class="bf">pi</b> for [mathjaxinline]\pi[/mathjaxinline]; use * to denote multiplication, i.e. [mathjaxinline]pi*r = \pi r[/mathjaxinline]; use [mathjaxinline]\wedge[/mathjaxinline] for powers, i.e. [mathjaxinline]r\wedge 3 = r^3[/mathjaxinline]; use / for division, i.e. [mathjaxinline]a/b = \frac{a}{b}[/mathjaxinline]) </p>
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<p style="display:inline">[mathjaxinline]A=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\displaystyle \frac{dA}{dr}=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\displaystyle \left.\frac{dA}{dr}\right|_{r=3}=[/mathjaxinline]</p>
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Area problem 2
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Suppose that I am building a hot tub. The area of the hot tub is important, because this determines how many people will fit in the tub. But building materials are expensive, so I want to know how the area of my hot tub changes with respect to the circumference of the hot tub. Rewrite the formula for the area in terms of the circumference [mathjaxinline]c[/mathjaxinline]. </p>
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(Type <b class="bf">pi</b> for [mathjaxinline]\pi[/mathjaxinline]; use * to denote multiplication, i.e. [mathjaxinline]pi*c = \pi c[/mathjaxinline]; use [mathjaxinline]\wedge[/mathjaxinline] for powers, i.e. [mathjaxinline]c\wedge 3 = c^3[/mathjaxinline]; use / for division, i.e. [mathjaxinline]a/b = \displaystyle \frac{a}{b}[/mathjaxinline]) </p>
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<p style="display:inline">[mathjaxinline]A=[/mathjaxinline]</p>
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<div class="hideshowcontent">You may want to use the formula [mathjaxinline]c = 2\pi r.[/mathjaxinline] Rearrange to find <em>r</em> in terms of <em>c</em>, and substitute into the formula for the area you found above. </div>
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Calculate the derivative. <p style="display:inline">[mathjaxinline]\displaystyle \frac{dA}{dc}=[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_der_5-tab4-problem2_3_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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Earlier we calculated [mathjaxinline]\displaystyle \frac{dA}{dr}[/mathjaxinline] when [mathjaxinline]r=3[/mathjaxinline]. When [mathjaxinline]r=3[/mathjaxinline], the circumference is [mathjaxinline]c=6\pi[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\left. \displaystyle \frac{dA}{dc}\right|_{c=6\pi } =[/mathjaxinline]</p>
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You probably got different answers for the answers for the values of [mathjaxinline]\displaystyle \left.\frac{dA}{dr}\right|_{r=3}[/mathjaxinline] and [mathjaxinline]\left. \displaystyle \frac{dA}{dc}\right|_{c=6\pi }[/mathjaxinline]. Try to think of why that might be. </p>
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<h2 class="hd hd-2 unit-title">6. Derivatives act on functions</h2>
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<ul class="itemize"><li><p><b class="bf">Units:</b> If [mathjaxinline]P[/mathjaxinline] has units of pressure, and [mathjaxinline]t[/mathjaxinline] has units of [mathjaxinline]time[/mathjaxinline], then [mathjaxinline]\displaystyle \frac{dP}{dt}[/mathjaxinline] has units of pressure per time. </p></li><li><p><b class="bf">Evaluating at points:</b> If we want to take the derivative at a particular point [mathjaxinline]x = 3[/mathjaxinline], then we use the notation [mathjaxinline]\left.\displaystyle \frac{df}{dx}\right|_{x=3}[/mathjaxinline]. The bar is read as “evaluated at". </p></li><li><p><b class="bf">Derivatives act on functions:</b></p><ul class="itemize"><li><p>
We can write [mathjaxinline]\displaystyle \frac{d(x^2)}{dx}[/mathjaxinline] for the derivative of [mathjaxinline]x^2[/mathjaxinline]. </p></li><li><p>
If a formula is long, we can write [mathjaxinline]\displaystyle \frac{d}{dy} \left(y^3 + 2y^2 \right).[/mathjaxinline] </p></li></ul></li></ul>
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Quick exercise using Leibniz notation
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What is [mathjaxinline]\displaystyle \frac{d}{dz} \left.\left( z^3 + 2z^2\right) \right|_{z=2}[/mathjaxinline]? </p>
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<h2 class="hd hd-2 unit-title">7. Review</h2>
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Suppose that the amount of water [mathjaxinline]W[/mathjaxinline] needed to grow tomatoes is a function of the kilograms of tomatoes [mathjaxinline]t[/mathjaxinline], given by the formula [mathjaxinline]W = f(t) = 10t^3-5t^2.[/mathjaxinline] Which of the following would be proper notation for the derivative of [mathjaxinline]W[/mathjaxinline] with respect to [mathjaxinline]t[/mathjaxinline]? (Select as many as will work.) </p>
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<text>[mathjaxinline]f'(t)[/mathjaxinline]</text>
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<text>[mathjaxinline]f'(W)[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \frac{df}{dx}[/mathjaxinline]</text>
</label>
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<input type="checkbox" name="input_der_5-tab7-problem1_2_1[]" id="input_der_5-tab7-problem1_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="der_5-tab7-problem1_2_1-choice_3-label" for="input_der_5-tab7-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_der_5-tab7-problem1_2_1">
<text>[mathjaxinline]\displaystyle \frac{dW}{dt}[/mathjaxinline]</text>
</label>
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<input type="checkbox" name="input_der_5-tab7-problem1_2_1[]" id="input_der_5-tab7-problem1_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="der_5-tab7-problem1_2_1-choice_4-label" for="input_der_5-tab7-problem1_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_der_5-tab7-problem1_2_1">
<text>[mathjaxinline]\displaystyle \frac{df}{dW}[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \frac{dt}{dW}[/mathjaxinline]</text>
</label>
</div>
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<input type="checkbox" name="input_der_5-tab7-problem1_2_1[]" id="input_der_5-tab7-problem1_2_1_choice_6" class="field-input input-checkbox" value="choice_6"/><label id="der_5-tab7-problem1_2_1-choice_6-label" for="input_der_5-tab7-problem1_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_der_5-tab7-problem1_2_1">
<text>[mathjaxinline]\displaystyle \frac{df}{dt}[/mathjaxinline]</text>
</label>
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<input type="checkbox" name="input_der_5-tab7-problem1_2_1[]" id="input_der_5-tab7-problem1_2_1_choice_7" class="field-input input-checkbox" value="choice_7"/><label id="der_5-tab7-problem1_2_1-choice_7-label" for="input_der_5-tab7-problem1_2_1_choice_7" class="response-label field-label label-inline" aria-describedby="status_der_5-tab7-problem1_2_1">
<text>[mathjaxinline]\displaystyle \frac{df}{dt}\left(10t^3-5t^2\right)[/mathjaxinline]</text>
</label>
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<input type="checkbox" name="input_der_5-tab7-problem1_2_1[]" id="input_der_5-tab7-problem1_2_1_choice_8" class="field-input input-checkbox" value="choice_8"/><label id="der_5-tab7-problem1_2_1-choice_8-label" for="input_der_5-tab7-problem1_2_1_choice_8" class="response-label field-label label-inline" aria-describedby="status_der_5-tab7-problem1_2_1">
<text>[mathjaxinline]\displaystyle \frac{d}{dt}\left(10t^3-5t^2\right)[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \frac{d}{dx}\left(10t^3-5t^2\right)[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">8. Summary</h2>
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<p><b class="bfseries">Why Leibniz notation?</b></p><p>
Why do we like to use Leibniz notation? </p><p>
The biggest reason is that it reminds us what the input variable is. The derivative is measuring the instantaneous rate of change of the output variable of a function with respect to the input variable. Sometimes, if there are lots of quantities that have variables representing them, it's easy to lose track of what is what; Leibniz notation helps to remind us. </p><p><b class="bfseries">Properties of Leibniz notation</b></p><ul class="itemize"><li><p><b class="bf">Units:</b> If [mathjaxinline]P[/mathjaxinline] has units of pressure, and [mathjaxinline]t[/mathjaxinline] has units of [mathjaxinline]time[/mathjaxinline], then [mathjaxinline]\displaystyle \frac{dP}{dt}[/mathjaxinline] has units of pressure per time. </p></li><li><p><b class="bf">Evaluating at points:</b> If we want to take the derivative at a particular point [mathjaxinline]x = 3[/mathjaxinline], then we use the notation [mathjaxinline]\left.\displaystyle \frac{df}{dx}\right|_{x=3}[/mathjaxinline]. The bar is read as “evaluated at". </p></li><li><p><b class="bf">Derivatives act on functions:</b></p><ul class="itemize"><li><p>
We can write [mathjaxinline]\displaystyle \frac{d(x^2)}{dx}[/mathjaxinline] for the derivative of [mathjaxinline]x^2[/mathjaxinline]. </p></li><li><p>
If a formula is long, we can write [mathjaxinline]\displaystyle \frac{d}{dy} \left(y^3 + 2y^2 \right).[/mathjaxinline] </p></li></ul></li></ul>
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