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You CAN Ace Calculus

17calculus > derivatives > exponential and logarithm

 basic derivative rules power rule product rule quotient rule For the basic exponential and logarithm 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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Exponential and Logarithm Derivatives

The derivatives of the exponential and the logarithm are very different. However, since the functions themselves are closely related, we discuss both of them on this page.

 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.

Okay, so what do you 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 $$e$$.
$$\displaystyle{ \begin{array}{rcl} y & = & a^x \\ \ln(y) & = & \ln(a^x) \\ & = & x\ln(a) \\ e^{\ln(y)} & = & e^{x\ln(a)} \\ y & = & e^{x\ln(a)} \end{array} }$$

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

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

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
 Exponential Practice

Unless otherwise instructed, calculate the derivative of these functions.

exponential practice filters

no chain rule

chain rule required

 Basic Problems

Practice 1

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

solution

Practice 2

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

solution

Practice 3

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

solution

Practice 4

$$\displaystyle{y=e^x(x+x\sqrt{x})}$$

solution

Practice 5

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

solution

Practice 6

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

solution

Practice 7

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

solution

 Intermediate Problems

Practice 8

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

solution

Practice 9

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

solution

Practice 10

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

solution

Practice 11

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

solution

Practice 12

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

solution

 Logarithm Derivative

The rule is not as simple as the exponential derivative but it is still very straightforward.

Basic Logarithm Rule

$$\displaystyle{ \frac{d}{dt}[\ln(t)] = \frac{1}{t} }$$

Logarithm With Chain Rule

$$\displaystyle{ \frac{d}{dt}[\ln(u)] = \frac{1}{u}\frac{du}{dt} }$$

It is probably not clear just from the equation that the derivative of $$\ln(x)$$ is $$1/x$$. Here is a great video explaining, first intuitively, then from the limit, where this derivative comes from. Although he says you can stop the video after the intuitive explanation, watching the entire video will help you a lot (it's not very long).

 MathTV - The Derivative of the Natural Log Function

Many times, it helps to simplify a logarithmic expression before taking the derivative. Here are a few rules that should help you.

 $$\ln(xy) = \ln(x) + \ln(y)$$ $$\ln(x/y) = \ln(x) - \ln(y)$$ $$\ln(x^y) = y \ln(x)$$ review logarithms

Here is a short video clip that goes through these equations again.

 PatrickJMT - Derivatives of Logarithmic Functions and Examples

Before jumping into some practice problems, take a couple of minutes to watch this next video. It will help you see some common mistakes that you can avoid when taking the derivative of logarithm functions.

 MathTV - Common Mistakes for Natural Logs and the Chain Rule
 Logarithm Practice

Unless otherwise instructed, calculate the derivative of these functions.

logarithm practice filters

no chain rule

chain rule required

 Basic Problems

Practice 13

$$\displaystyle{f(x)=\frac{1}{\ln(x)}}$$

solution

Practice 14

$$\displaystyle{f(x)=\sqrt{x}\ln(x)}$$

solution

Practice 15

$$f(x)=\ln(x^2+10)$$

solution

Practice 16

$$y=\ln(x^2+x)$$

solution

Practice 17

$$\ln(3x^2+9x-5)$$

solution

Practice 18

$$h(x)=\ln(x^2+3x+4)$$

solution

Practice 19

$$f(x)=\ln(5x^2+2x-7)$$

solution

Practice 20

$$\displaystyle{g(x)=\log_4(x^3+8x)}$$

solution

Practice 21

$$\displaystyle{f(x)=\ln\sqrt[3]{x^3-x}}$$

solution

Practice 22

$$\displaystyle{f(x)=\ln\left(x\sqrt{x^2+1}\right)}$$

solution

Practice 23

$$y = \ln(x) + 2^x + \sin(x)$$

solution

Practice 24

Calculate the first three derivatives of $$f(x)=\ln(2+3x)$$.

solution

 Intermediate Problems

Practice 25

$$\displaystyle{y=\frac{\ln x}{1+\ln(2x)}}$$

solution

Practice 26

$$\displaystyle{f(x)=\ln\left[\frac{(2x+1)^5}{\sqrt{x^2+1}}\right]}$$

solution

Practice 27

$$\displaystyle{f(x)=\ln\left[\frac{(2x+1)^3}{(3x-1)^4}\right]}$$

solution

Practice 28

$$\displaystyle{y=\sqrt[3]{\log_7(x)}}$$

solution

Practice 29

$$y=\ln(x^4\sin x)$$

solution

Practice 30

$$\displaystyle{y=[\log_4(1+e^x)]^2}$$

solution

Practice 31

$$\displaystyle{p(x)=\ln\left(x^2\cdot\sqrt{x^3+3x}\cdot(x+2)^4\right)}$$

solution