You CAN Ace Calculus

Topics You Need To Understand For This Page

basic derivative rules

power rule

product rule

quotient rule

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.

Related Topics and Links

17Calculus Subjects Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

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.

learning and study techniques

Exponential Derivative

This is one of the easiest rules you will learn.

Basic Exponential Rule

\(\displaystyle{ \frac{d}{dt}[e^t] = e^t }\)

Exponential With Chain Rule

\(\displaystyle{ \frac{d}{dt}[e^u] = e^u \frac{du}{dt} }\)

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.

3Blue1Brown - Derivatives of exponentials [13min-49secs]

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.

PatrickJMT - Derivatives of Exponential Functions [5min-49secs]

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.

Dr Chris Tisdell - Derivative of exponentials [9min-15secs]

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.

MathTV - Some Natural Exponential Functions and Tangent Lines [4min-11secs]

video by MathTV

Practice

Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems

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.

GOT IT. THANKS!

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

Calculate the derivative of \(y=e^x(x+x\sqrt{x})\).

Solution

1069 solution video

video by Krista King Math

close solution

\(f(x)=4^x+3e^x+x^4\)

Problem Statement

Calculate the derivative of \(f(x)=4^x+3e^x+x^4\).

Solution

1074 solution video

video by PatrickJMT

close solution

\(f(x)=e^xx^2\)

Problem Statement

Calculate the derivative of \(f(x)=e^xx^2\).

Solution

1076 solution video

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

\(f(x)=(x^2-1)e^{-x}\)

Problem Statement

Calculate the derivative of \(f(x)=(x^2-1)e^{-x}\).

Solution

1066 solution video

video by Krista King Math

close solution

\(\displaystyle{f(x)=xe^{\sqrt{x}}}\)

Problem Statement

Calculate the derivative of \(\displaystyle{f(x)=xe^{\sqrt{x}}}\) .

Solution

1067 solution video

video by Krista King Math

close solution

\(\displaystyle{f(x)=\frac{1-e^{-x}}{x}}\)

Problem Statement

Calculate the derivative of \(\displaystyle{f(x)=\frac{1-e^{-x}}{x}}\).

Solution

1068 solution video

video by Krista King Math

close solution

\(\displaystyle{  3e^{ x^2+7 }  }\)

Problem Statement

Use the chain rule to calculate the derivative of \(\displaystyle{  3e^{ x^2+7 }  }\).

Final Answer

\(\displaystyle{ \frac{d}{dx} \left[ 3e^{ x^2+7 } \right] = 6xe^{x^2+7} }\)

Problem Statement

Use the chain rule to calculate the derivative of \(\displaystyle{  3e^{ x^2+7 }  }\).

Solution

\(\displaystyle{ \frac{d}{dx} \left[ 3e^{ x^2+7 } \right] }\)

\(\displaystyle{ 3e^{x^2+7} \cdot \frac{d}{dx}[x^2+7] }\)

\( 3e^{x^2+7} \cdot (2x) \)

\( 6xe^{x^2+7} \)

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

\(\displaystyle{ \frac{d}{dx} \left[ 3e^{ x^2+7 } \right] = 6xe^{x^2+7} }\)

close solution

Intermediate Problems

\(\displaystyle{f(x)=e^{x\sin(2x)}}\)

Problem Statement

Calculate the derivative of \(\displaystyle{f(x)=e^{x\sin(2x)}}\).

Solution

1077 solution video

video by PatrickJMT

close solution

\(\displaystyle{g(x)=2e^{\cos(x)\sin(5x)}}\)

Problem Statement

Calculate the derivative of \(\displaystyle{g(x)=2e^{\cos(x)\sin(5x)}}\).

Solution

1075 solution video

video by PatrickJMT

close solution

\(\displaystyle{f(t)=\cos\left(2^{\pi t}\right)}\)

Problem Statement

Calculate the derivative of \(\displaystyle{f(t)=\cos\left(2^{\pi t}\right)}\).

Solution

1078 solution video

video by PatrickJMT

close solution

For what values of x does \(h(x)=5e^{5x}-25x\) have negative derivatives?

Problem Statement

For what values of x does \(h(x)=5e^{5x}-25x\) have negative derivatives?

Solution

1079 solution video

video by PatrickJMT

close solution

\(\displaystyle{ y = \cos \left( \frac{1-e^{2x}}{1+e^{2x}} \right) }\)

Problem Statement

Use the chain rule to calculate the derivative of \(\displaystyle{ y = \cos \left( \frac{1-e^{2x}}{1+e^{2x}} \right) }\).

Solution

986 solution video

video by Krista King Math

close solution
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