Graphing a Derivative

We have already discussed how to graph a function, so given the equation of a function or the equation of a derivative function, we could graph it. Given both, we would expect to see a correspondence between the graphs of these two functions, since \(f^{'} \left(\right. x \left.\right)\) gives the rate of change of a function \(f \left(\right. x \left.\right)\) (or slope of the tangent line to \(f \left(\right. x \left.\right) ).\)

In Example 3.11 we found that for \(f \left(\right. x \left.\right) = \sqrt{x} , f^{'} \left(\right. x \left.\right) = \frac{1}{2 \sqrt{x}} .\) If we graph these functions on the same axes, as in Figure 3.12, we can use the graphs to understand the relationship between these two functions. First, we notice that \(f \left(\right. x \left.\right)\) is increasing over its entire domain, which means that the slopes of its tangent lines at all points are positive. Consequently, we expect \(f^{'} \left(\right. x \left.\right) > 0\) for all values of \(x\) in its domain. Furthermore, as \(x\) increases, the slopes of the tangent lines to \(f \left(\right. x \left.\right)\) are decreasing and we expect to see a corresponding decrease in \(f^{'} \left(\right. x \left.\right) .\) We also observe that \(f ' \left(\right. 0 \left.\right)\) is undefined and that \(\underset{x \rightarrow 0^{+}}{\text{lim}} f^{'} \left(\right. x \left.\right) = + \infty ,\) corresponding to a vertical tangent to \(f \left(\right. x \left.\right)\) at \(0 .\)

Figure 3.12 The derivative \(f^{'} \left(\right. x \left.\right)\) is positive everywhere because the function \(f \left(\right. x \left.\right)\) is increasing.

In Example 3.12 we found that for \(f \left(\right. x \left.\right) = x^{2} - 2 x , f^{'} \left(\right. x \left.\right) = 2 x - 2 .\) The graphs of these functions are shown in Figure 3.13. Observe that \(f \left(\right. x \left.\right)\) is decreasing for \(x < 1 .\) For these same values of \(x , f^{'} \left(\right. x \left.\right) < 0 .\) For values of \(x > 1 , f \left(\right. x \left.\right)\) is increasing and \(f^{'} \left(\right. x \left.\right) > 0 .\) Also, \(f \left(\right. x \left.\right)\) has a horizontal tangent at \(x = 1\) and \(f^{'} \left(\right. 1 \left.\right) = 0 .\)

Figure 3.13 The derivative \(f^{'} \left(\right. x \left.\right) < 0\) where the function \(f \left(\right. x \left.\right)\) is decreasing and \(f^{'} \left(\right. x \left.\right) > 0\) where \(f \left(\right. x \left.\right)\) is increasing. The derivative is zero where the function has a horizontal tangent.

Example 3.13

Sketching a Derivative Using a Function

Use the following graph of \(f \left(\right. x \left.\right)\) to sketch a graph of \(f^{'} \left(\right. x \left.\right) .\)

Solution

The solution is shown in the following graph. Observe that \(f \left(\right. x \left.\right)\) is increasing and \(f^{'} \left(\right. x \left.\right) > 0\) on \(\left(\right. – 2 , 3 \left.\right) .\) Also, \(f \left(\right. x \left.\right)\) is decreasing and \(f^{'} \left(\right. x \left.\right) < 0\) on \(\left(\right. − \infty , −2 \left.\right)\) and on \(\left(\right. 3 , + \infty \left.\right) .\) Also note that \(f \left(\right. x \left.\right)\) has horizontal tangents at \(– 2\) and \(3 ,\) and \(f^{'} \left(\right. −2 \left.\right) = 0\) and \(f^{'} \left(\right. 3 \left.\right) = 0 .\)