Mathematics 265 Introduction to Calculus I

Study Guide :: Unit 4

Differentiation

The Graph of the Derivative Function

Observe that of the two forms for $f^{\prime }\left(a\right)$, that is,

${\displaystyle f^{\prime }\left(a\right)=\underset{h\to 0}{\lim }{\displaystyle \frac{f\left(a+h\right)-f\left(a\right)}{h}}\text{and}{f}^{\prime }\left(a\right)=\underset{x\to a}{\lim }{\displaystyle \frac{f\left(x\right)-f\left(a\right)}{x-a}}\text{,}}$

we can replace the number $a$ with $x$ only in the former. Doing so gives us

${\displaystyle f^{\prime }\left(x\right)=\underset{h\to 0}{\lim }{\displaystyle \frac{f\left(x+h\right)-f\left(x\right)}{h}}\text{.}}$

This general form of $f'(x)$ for any number $x$ gives a function that is called the “derivative function of $f$.”

The domain of the derivative function is the set of all $x$ for which the limit exists; that is, the set of numbers for which the function $f$ is differentiable.

The value of $f'(x)$ has two interpretations:

1. it is the instantaneous rate of change at $x$.
2. it is the slope of the tangent line at the point $\left(x,f\left(x\right)\right)$.

Using the geometric interpretation of the derivative function, we can sketch its graph by observing closely how the slope of the tangent line changes as it is “travels” through the curve.

Examine the graph of the function given in Figure 4.7, below, and pay attention to the slopes $S_1$, $S_2$, $S_3$, $S_4$, and $S_5$, of the tangent lines. As you can see, all the tangent lines are positive; furthermore,

$S_1>S_2>S_3\text{and}{S}_3<S_4<S_5\text{.}$

Figure 4.7. Function showing positive slopes of the tangent lines

As the tangent lines “travel” through the curve, the slopes decrease before $a$ and increase after $a$. Since the derivative function is positive, decreasing before $a$ and increasing after $a$, its graph might look like that shown in Figure 4.8, below.

Figure 4.8. Possible derivative function for the function in Figure 4.7

In Figure 4.9, below, as the tangent line travels through the curve, the slopes $S_1\text{,}{S}_2\text{,}{S}_3\text{,}{S}_4$ and $S_5$ are negative; furthermore,

$S_1<S_2<S_3\text{and}{S}_3>S_4>S_5\text{.}$

Figure 4.9. Function showing negative slopes of the tangent lines

Thus, the graph of the derivative function is increasing before $a$ and decreasing after $a$, and its graph might look like that shown in Figure 4.10, below.

Figure 4.10. Possible derivative function for the function in Figure 4.9

If the tangent line is horizontal, then the slope is zero, and the derivative function is zero at this point. In Figure 4.11, below, the slopes $S_1\text{,}{S}_2\text{,}{S}_3\text{,}{S}_4$ and $S_5$ are negative, and

$S_1<S_2<S_3<S_4<S_5=0$

Figure 4.11. Function with negative tangents and one horizontal tangent

In Figure 4.12, below, the slopes $S_6\text{,}{S}_7\text{,}{S}_8$ and $S_9$ are positive, and

$0=S_5<S_6<S_7<S_8<S_9\text{.}$

Figure 4.12. Function with positive tangents and one horizontal tangent

The graph of the derivative function might be like that shown in Figure 4.13, below.

Figure 4.13. Possible derivative function for the function in Figure 4.12

To sketch the graph of the derivative function $f'(x)$ from the graph of the function $f(x)$:

1. identify the points where the derivative is zero; that is, where the tangent line is horizontal.
2. identify the points where the derivative is undefined; that is, where there is no tangent line or where the tangent line is vertical.
3. identify the intervals where the derivative is positive or negative.
4. identify the intervals where the derivative is increasing or decreasing.

Example 4.17. In Figure 4.14, below, we have the graph of a function $f\left(x\right)\text{.}$

Figure 4.14. Graph of a function $f(x)$, Example 4.17

To sketch the graph of the derivative function, $f'(x)$, we must note that

1. $f^{\prime }\left(a\right)=0$ and $f^{\prime }\left(c\right)=0$, because at the points $\left(a,f\left(a\right)\right)$ and $\left(c,f\left(c\right)\right)$, the tangent lines are horizontal.
2. $f^{\prime }\left(b\right)$ is undefined, because there is no tangent line at the point $\left(b,f\left(b\right)\right)\text{.}$
3. $f^{\prime }\left(x\right)>0$ on the interval $\left(a,b\right)$, because the tangent line on this interval is positive; and $f^{\prime }\left(x\right)<0$ on the intervals $\left(-\infty ,a\right)$, $\left(b,c\right)$, and $\left(c,\infty \right)$. [Why?]
4. $f'(x)$ is increasing on the intervals $\left(-\infty ,a\right)$, $\left(a,b\right)$ and $\left(b,c\right)$. [Why?]
5. $f'(x)$ is decreasing on the interval $\left(c,\infty \right)\text{.}$

A possible graph of the derivative function is shown in Figure 4.15 below. Later, we will learn to sketch this graph more precisely.

Figure 4.15. Possible derivative function $f'(x)$ for the function in Figure 4.14