## 17Calculus Derivatives - Exponential Functions

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.

video by MathTV

### Practice

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 video

video by Krista King Math

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

Problem Statement

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

Solution

### 1074 video

video by PatrickJMT

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

Problem Statement

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

Solution

### 1076 video

video by PatrickJMT

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 video

video by Krista King Math

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

Problem Statement

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

Solution

### 1067 video

video by Krista King Math

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

Problem Statement

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

Solution

### 1068 video

video by Krista King Math

$$\displaystyle{ 3e^{ x^2+7 } }$$

Problem Statement

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

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

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

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

Intermediate Problems

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

Problem Statement

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

Solution

### 1077 video

video by PatrickJMT

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

Problem Statement

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

Solution

### 1075 video

video by PatrickJMT

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

Problem Statement

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

Solution

### 1078 video

video by PatrickJMT

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 video

video by PatrickJMT

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

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 video

video by Krista King Math

You CAN Ace Calculus

 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.

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