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:

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.

Quiz

Take a quiz.

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.

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"

Information on Leibniz
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
Surfing Derivatives
Wolfram Research

Interesting Application

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

2.6 Tangent Lines and Their Slopes Table of Contents 2.8 Differentiation Rules


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