The Derivative as a Function: Learn It 2

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 f(x) gives the rate of change of a function f(x) or the slope of the tangent line to f(x). 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 0.
  • Undefined Slope: A vertical line has an undefined slope.

In the first example on the previous page, we found that for f(x)=x,f(x)=12x. 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.

The function f(x) = the square root of x is graphed as is its derivative f’(x) = 1/(2 times the square root of x).
Figure 2. The derivative f(x) is positive everywhere because the function f(x) is increasing.

Looking at the graphs, notice that f(x) is increasing over its entire domain, meaning the slopes of its tangent lines at all points are positive. Consequently, f(x)>0 for all values of x in its domain. As x increases, the slopes of the tangent lines to f(x) decrease, leading to a corresponding decrease in f(x). Additionally, f(0) is undefined and that limx0+f(x)=+, corresponding to a vertical tangent to f(x) at 0.

In the second example, we found that for f(x)=x22x,f(x)=2x2. The graphs of these functions are shown in Figure 3. 

The function f(x) = x squared – 2x is graphed as is its derivative f’(x) = 2x − 2.
Figure 3. The derivative f(x)<0 where the function f(x) is decreasing and f(x)>0 where f(x) is increasing. The derivative is zero where the function has a horizontal tangent.

Observe that f(x) is decreasing for x<1. For these values of x,f(x)<0. For x>1,f(x) is increasing and f(x)>0. Also, f(x) has a horizontal tangent at x=1 and f(1)=0.

Use the following graph of f(x) to sketch a graph of f(x).

The function f(x) is roughly sinusoidal, starting at (−4, 3), decreasing to a local minimum at (−2, 2), then increasing to a local maximum at (3, 6), and getting cut off at (7, 2).
Figure 4. Graph of f(x).