Derivatives -- Comparing a Function with Its Derivative

You should use one of the computer algebra systems below with this module. Click on the appropriate icon for your preferred CAS and then arrange your screen so that you can easily move back-and-forth between this window and your CAS window. Click on the appropriate help button for help.

If we start with a function f(x) then at each point x we can find the slope of the tangent to the curve y = f(x) at the point (x, f(x)). The slope of the tangent is given by

                 f(s) - f(x)
f'(x) =   lim    ----------- 
        s --> x     s - x

This gives us a new function, f'(x), called the derivative of the original function. The movie below illustrates this idea. In the first frame we see the graph of a function f(x). As the movie plays we slide a tangent along the curve from left to right and as the tangent slides we draw the graph of the new function y = f'(x) (in blue).

Missing movie Graphing the Derivative of a Function

Notice the following about this movie.

At each point x at which the function f(x) is decreasing the slope of the tangent is negative -- that is, f'(x) is negative.
At each point x at which the function f(x) is increasing the slope of the tangent is positive -- that is, f'(x) is positive.
At each point x which is the top of a mountain or the bottom of a valley the tangent is horizontal, so its slope is zero -- that is, f'(x) = 0.

Check Your Understanding

For each of the following functions do the following.

Use your CAS window to draw a graph of the function and a graph of its derivative on the same set of axes.
Determine which of the two graphs is the function and which is the derivative.

f(x) = sin x
answer
f(x) = x^2
answer
f(x) = x^3
answer
f(x) = x^3 - x
answer
f(x) = 1/x
answer
f(x) = 4x
answer