You CAN Ace Calculus
For the basic exponential derivatives you do not need the chain rule. But we discuss it on this page. Each section is labeled. So if you have not studied the chain rule yet, you can read the sections that apply to you and then come back here once you have studied it. |
external links you may find helpful |
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WikiBooks - Derivatives of Exponential and Logarithm Functions |
Single Variable Calculus |
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Multi-Variable Calculus |
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Acceleration Vector |
Arc Length (Vector Functions) |
Arc Length Function |
Arc Length Parameter |
Conservative Vector Fields |
Cross Product |
Curl |
Curvature |
Cylindrical Coordinates |
Lagrange Multipliers |
Line Integrals |
Partial Derivatives |
Partial Integrals |
Path Integrals |
Potential Functions |
Principal Unit Normal Vector |
Differential Equations |
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Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.
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Help Keep 17Calculus Free |
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Exponential Derivative |
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This is one of the easiest rules you will learn.
Basic Exponential Rule |
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\(\displaystyle{ \frac{d}{dt}[e^t] = e^t }\) |
Exponential With Chain Rule |
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\(\displaystyle{ \frac{d}{dt}[e^u] = e^u \frac{du}{dt} }\) |
It looks like we didn't do anything here. However, the exponential function is the only function whose derivative is itself.
Before we go on, let's watch a video that gives an intuitive explanation of the derivative of exponential functions and why \(f(x)=e^x\) is special.
video by 3Blue1Brown
Okay, so what do you do if you have a base other than \(e\)? The formula is fairly straightforward but let's derive from our rules of logarithms. Since we don't like to memorize formulas and we already know the logarithm rules, why not just derive it when we need it, since we don't use it very much?
So, let's convert \(y=a^x\), where \(a\) is a constant, into a form with \(e\).
\( \begin{array}{rcl}
y & = & a^x \\
\ln(y) & = & \ln(a^x) \\
& = & x\ln(a) \\
e^{\ln(y)} & = & e^{x\ln(a)} \\
y & = & e^{x\ln(a)}
\end{array} \)
So now when we take the derivative of \(y = a^x\), we can actually take the derivative of \(y=e^{x\ln(a)}\). Using the chain rule, we have \((a^x)' = (\ln(a))e^{x\ln(a)} \). Notice how didn't have to memorize this formula. We used the logarithm rules we already know.
This next video goes through all the explanation again. It is always good to get explanations from different sources since it will help you understand the material better.
video by PatrickJMT
Here is an interesting video that shows how to get the equation for the derivative of \(f(x)=a^x\) another way. He shows that \(\displaystyle{f'(x)=\frac{d[a^x]}{dx}=a^x f'(0)}\). This is an interesting and unusual way to think about the derivative.
video by Dr Chris Tisdell
So far, we've only been looking at equations with exponential functions. Here is a video discussing the graph, the derivative and the tangent line of three exponential functions. This helps you get more of an intuitive feel for this function and it's derivative.
video by MathTV
Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems |
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Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. Here is a list converting the old numbering system to the new. |
Exponential Derivatives - Practice Problems Conversion |
[1-1066] - [2-1067] - [3-1068] - [4-1069] - [5-1074] - [6-1076] - [7-973] - [8-1075] - [9-1077] |
[10-1078] - [11-986] - [12-1079] |
Please update your notes to this new numbering system. The display of this conversion information is temporary. |
Unless otherwise instructed, calculate the derivative of these functions. Here are a few practice problems that do not require the chain rule.
\(y=e^x(x+x\sqrt{x})\)
Problem Statement |
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Calculate the derivative of \(y=e^x(x+x\sqrt{x})\).
Solution |
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video by Krista King Math
close solution |
\(f(x)=4^x+3e^x+x^4\)
Problem Statement |
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Calculate the derivative of \(f(x)=4^x+3e^x+x^4\).
Solution |
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video by PatrickJMT
close solution |
\(f(x)=e^xx^2\)
Problem Statement |
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Calculate the derivative of \(f(x)=e^xx^2\).
Solution |
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video by PatrickJMT
close solution |
These practice problems require the chain rule. As before calculate the derivative of these functions, unless otherwise instructed.
Basic Problems |
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\(f(x)=(x^2-1)e^{-x}\)
Problem Statement |
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Calculate the derivative of \(f(x)=(x^2-1)e^{-x}\).
Solution |
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video by Krista King Math
close solution |
\(\displaystyle{f(x)=xe^{\sqrt{x}}}\)
Problem Statement |
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Calculate the derivative of \(\displaystyle{f(x)=xe^{\sqrt{x}}}\) .
Solution |
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video by Krista King Math
close solution |
\(\displaystyle{f(x)=\frac{1-e^{-x}}{x}}\)
Problem Statement |
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Calculate the derivative of \(\displaystyle{f(x)=\frac{1-e^{-x}}{x}}\).
Solution |
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video by Krista King Math
close solution |
\(\displaystyle{ 3e^{ x^2+7 } }\)
Problem Statement |
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Use the chain rule to calculate the derivative of \(\displaystyle{ 3e^{ x^2+7 } }\).
Final Answer |
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\(\displaystyle{ \frac{d}{dx} \left[ 3e^{ x^2+7 } \right] = 6xe^{x^2+7} }\) |
Problem Statement |
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Use the chain rule to calculate the derivative of \(\displaystyle{ 3e^{ x^2+7 } }\).
Solution |
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\(\displaystyle{ \frac{d}{dx} \left[ 3e^{ x^2+7 } \right] }\) |
\(\displaystyle{ 3e^{x^2+7} \cdot \frac{d}{dx}[x^2+7] }\) |
\( 3e^{x^2+7} \cdot (2x) \) |
\( 6xe^{x^2+7} \) |
Another way to work this is with the substitution method.
let \(u=x^2+7\) |
\(\displaystyle{ \frac{d}{dx} \left[ 3e^{ x^2+7 } \right] }\) |
\(\displaystyle{ 3\frac{d[e^u]}{du} \cdot \frac{d[x^2+7]}{dx} }\) |
\( 3e^u \cdot (2x) \) |
\( 6xe^{x^2+7} \) |
Final Answer |
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\(\displaystyle{ \frac{d}{dx} \left[ 3e^{ x^2+7 } \right] = 6xe^{x^2+7} }\) |
close solution |
Intermediate Problems |
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\(\displaystyle{f(x)=e^{x\sin(2x)}}\)
Problem Statement |
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Calculate the derivative of \(\displaystyle{f(x)=e^{x\sin(2x)}}\).
Solution |
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video by PatrickJMT
close solution |
\(\displaystyle{g(x)=2e^{\cos(x)\sin(5x)}}\)
Problem Statement |
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Calculate the derivative of \(\displaystyle{g(x)=2e^{\cos(x)\sin(5x)}}\).
Solution |
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video by PatrickJMT
close solution |
\(\displaystyle{f(t)=\cos\left(2^{\pi t}\right)}\)
Problem Statement |
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Calculate the derivative of \(\displaystyle{f(t)=\cos\left(2^{\pi t}\right)}\).
Solution |
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video by PatrickJMT
close solution |
For what values of x does \(h(x)=5e^{5x}-25x\) have negative derivatives?
Problem Statement |
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For what values of x does \(h(x)=5e^{5x}-25x\) have negative derivatives?
Solution |
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video by PatrickJMT
close solution |
\(\displaystyle{ y = \cos \left( \frac{1-e^{2x}}{1+e^{2x}} \right) }\)
Problem Statement |
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Use the chain rule to calculate the derivative of \(\displaystyle{ y = \cos \left( \frac{1-e^{2x}}{1+e^{2x}} \right) }\).
Solution |
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video by Krista King Math
close solution |