## 2.7 The Derivative

### Summary

The derivative of a function is defined. The derivative as a function in its own right is also discussed. Furthermore, derivatives are computed both graphically and from the limit definition, and the power rule is developed.

By the end of your studying, you should know:

• The limit definition of the derivative.
• How to recognize properties of the derivative from a graph of a function.
• Rules for finding the derivative of a linear function, and a function that is a power of x.
• The Power Rule.
• The derivative of |x| and 1/x.
• Different notations for the derivative.
• How to compute higher derivatives.

On-screen applet instructions: Use the slider to control the value of h. Note that the number of difference quotients computed and plotted increases as h->0. The button under the slider shows and hides the derivative curve. Whenever you like, you can compare a computed set of difference quotients with the derivative. What is the geometrical significance of the limit of the difference quotient as h -> 0?

### Examples

Let f(x) = x2 and g(x) = x. Find (f + g)' (4). Does this equal f '(4) + g'(4)?

Draw the derivative of the following graph.

Let g(x) = 1/x. Find g'(x), g''(x), and g'''(x), and graph them. Can you find a formula for the nth derivative of g(x)?

### Videos

See short videos of worked problems for this section.

### Exercises

See Exercises for 2.7 The Derivative (PDF).

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

#### Interesting Application

Average reaction rates and instantaneous reaction rates in chemistry can be quite different.

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