To begin, we recall two basic facts about the derivative $f'(x)$ of a function $f(x):$

1. The value $f'(a)$ of $f'(x)$ at $x = a$ is the slope of the tangent to the graph of the function $f$ at the point where $x = a.$
2. $f'(x)$ is a function of $x:$ the slope at a point on the graph depends on the $x$-coordinate of that point.

The graph of the derivative function $f'(x)$ gives us interesting information about the original function $f(x).$ The following example shows us how to sketch the graph of $f'(x)$ from a knowledge of the graph of $f(x).$
_{} Example 1 Sketching the Graph of the Derivative
Let $f(x)$ have the graph shown below.

Give a rough sketch of the graph of $f'(x).$

Solution

Remember that $f'(x)$ is the slope of the tangent at the point $(x, f(x))$ on the graph of $f.$ To sketch the graph of $f',$ we make a table with several values of $x$ (the corresponding points are shown on the graph) and rough estimates of the slope of the tangent $f'(x).$

$x$

$0$

$0.5$

$1$

$1.5$

$2$

$2.5$

$3$

$f'(x)$

$3$

$0$

$-4$

$-3$

$0$

$1$

$0$

(Note that rough estimates are the best we can do; it is difficult to measure the slope of the tangent accurately without using a grid and a ruler, so we couldn't reasonably expect two people's estimates to agree. However, all that is asked for is a rough sketch of the derivative.) Plotting these points suggests the curve shown below.

Notice that the graph $f'(x)$ intersects the $x$-axis at points that correspond to the high and low points on the graph of $f(x).$ Why is this so?

Here is a more interactive example.

_{} Example 2 Graph of Derivative
Let $f(x)$ have the graph shown below.

Complete the following table, giving rough estimates of the slope of the tangent $f'(x)$ at the given values of $x.$

Now plot these points, and hence make a rough sketch of the graph of $f'(x).$ Which of the following best approximates your sketch of the graph of $f'(x)$? (click on one)