## 2.12 Derivatives of Exponential and Logarithm Functions

### Summary

The derivatives of the exponential and logarithm functions are computed.

By the end of your studying, you should know:

• The derivative of ex.
• The derivative of ln(x).
• How to write ax in terms of ex; and to use this formula to compute the derivative of ax.
• How to write log to the base a of x in terms of ln(x); and to use this formula to compute the derivative of the general logarithm function.
• How to find a solution of the differential equation y' = ky.

On-screen applet instructions: For any x, the red dot represents the difference quotient for the natural log at x for a given h. Use the slider to let h → 0.

### Examples

Let y = ln(x2 + y2). Find y'.

Let

Find y'.

Show that there is a point at which the derivative of the function p(x) = ex – 3x is equal to zero.

### Applets

Limits of Functions

### Videos

See short videos of worked problems for this section.

### Exercises

See Exercises for 2.12 Derivatives of Exponential and Logarithm Functions (PDF).

Work online to solve the exercises for this section, or for any other section of the textbook.

### Resources on the Web

Information on Newton
Biographical data from St. Andrew's University's Web site
Excerpt from W.W. Rouse Ball's "A Short Account of the History of Mathematics"

Calculus Applications
Project Intermath

Derivatives of Exponentials and Logs
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Copyright © 2005 Donald L. Kreider, C. Dwight Lahr, Susan J. Diesel