Introduction to the Derivative 3

3.5 The Derivative: Numerical and Graphical Viewpoints

First tutorial: Part A: Numerical Approach

Second tutorial: Part B: Graphical Viewpoint

This tutorial: Part C: The Derivative Function

(This topic is also in Section 3.5 in Applied Calculus or Section 10.5 in Finite Mathematics and Applied Calculus))

The derivative f'(x) is a number we can calculate, or at least approximate, for various values of x. Since f'(x) depends on the value of x, we may think of f' as a function of x. This function is the derivative function.

Derivative Function

If f is a function, its derivative function f' is the function whose value f'(x) at x is the derivative of f at x. Its domain is the set of all x at which f is differentiable. Equivalently, f' associates to each x the slope of the tangent to the graph of the function f at x, or the instantaneous rate of change of f at x. Thus,

f'(x)

Slope of Tangent at x

lim
h→0

f(x+h) - f(x)

In short: The derivative function is the slope function -- its value at x gives the slope of the function at x. Put another way, the graph of the derivative function shows the slope of the original function.

Example
Here is an illutration of the derivative function. At each value of x, the derivative function f' gives the slope of the tangent line at x. Click on the buttons to see how this function depends on x. (Each gridline represents one unit.)

Let f have the graph shown:

Which of the following is the graph of the derivative, f'? (Click on the correct graph.) [Remember: the graph of the derivative gives the slope of the original function.]

To download an Excel utility that plots the derivative of any function you enter, click here.
Note: Make sure that macros as "enabled" or else the utility will not work.

You could now try some of the exercises in Section 3.5 in Applied Calculus or Section 10.5 in Finite Mathematics and Applied Calculus.

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