Graphing a Derivative
Given the equation of a function or its derivative, we can graph it to understand the relationship between the two. The derivative gives the rate of change of a function or the slope of the tangent line to . To understand this relationship better, it is helpful to recall the characteristics of lines with certain slopes.
- Positive Slope: A line has a positive slope if it is increasing from left to right.
- Negative Slope: A line has a negative slope if it is decreasing from left to right.
- Zero Slope: A horizontal line has a slope of .
- Undefined Slope: A vertical line has an undefined slope.
In the first example on the previous page, we found that for . If we graph these functions on the same axes, as in Figure 2, we can use the graphs to understand the relationship between these two functions.

Looking at the graphs, notice that is increasing over its entire domain, meaning the slopes of its tangent lines at all points are positive. Consequently, for all values of in its domain. As increases, the slopes of the tangent lines to decrease, leading to a corresponding decrease in . Additionally, is undefined and that , corresponding to a vertical tangent to at .
In the second example, we found that for . The graphs of these functions are shown in Figure 3.

Observe that is decreasing for . For these values of . For is increasing and . Also, has a horizontal tangent at and .
Use the following graph of to sketch a graph of .
