## 2.8 Differentiation Rules

### Summary

If there had not been easily applied rules for finding the derivative of most functions used in modeling, the derivative would not be as powerful a tool as it has turned out to be. Using the limit definition, general rules are developed for constant multiples of a function; sums, products, reciprocals, quotients, and compositions of functions. Along with the Power Rule, these rules permit the calculation of the derivative of a remarkably large number of functions.

By the end of your studying, you should know:

• Differentiability implies continuity, and an example showing that the converse is false.
• Constant multiple rule for derivatives.
• The Sum Rule (for finding the derivative of a sum of functions).
• The Product Rule.
• The Reciprocal Rule.
• The Quotient Rule.
• The Chain Rule.

On-screen applet instructions: The table displayed shows the values of the difference quotients of fg for the given value of h and at various points x. The value of h may be controlled by the slider so that one can investigate the limit of the difference quotients as h → 0. The hide/show button at the bottom will show an additional column of values for comparison with the derivative fg′ + f′g.

### Examples

Differentiate h(x) = (x + 2)4.

Robert throws a rock into a lake, which creates a circle of ripples which moves away from the point of impact at a constant speed of 50 centimeters per second. What is the rate of change in the area of the circle after 1 second? After 10 seconds? After t seconds?

Find the instantaneous rate of change of the volume of a cube with respect to the length of its edge, x, when x equals 4 inches.

### Videos

See short videos of worked problems for this section.

### Exercises

See Exercises for 2.8 Differentiation Rules (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

Table of Differentiation Rules
World Web Math

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Copyright © 2005 Donald L. Kreider, C. Dwight Lahr, Susan J. Diesel