The graphs of two differentiable functions f and g are shown above given p x = f(x g(x))

As we have seen, the derivative of a function at a given point gives us the rate of change or slope of the tangent line to the function at that point. If we differentiate a position function at a given time, we obtain the velocity at that time. It seems reasonable to conclude that knowing the derivative of the function at every point would produce valuable information about the behavior of the function. However, the process of finding the derivative at even a handful of values using the techniques of the preceding section would quickly become quite tedious. In this section we define the derivative function and learn a process for finding it.

The derivative function gives the derivative of a function at each point in the domain of the original function for which the derivative is defined. We can formally define a derivative function as follows.

Definition

Let

be a function. The derivative function, denoted by

, is the function whose domain consists of those values of such that the following limit exists:

A function

is said to be differentiable at if

exists. More generally, a function is said to be differentiable on
if it is differentiable at every point in an open set

, and a differentiable function is one in which

exists on its domain.

In the next few examples we use (Figure) to find the derivative of a function.

Find the derivative of

Start directly with the definition of the derivative function. Use (Figure).

Find the derivative of the function

Follow the same procedure here, but without having to multiply by the conjugate.

Find the derivative of

We use a variety of different notations to express the derivative of a function. In (Figure) we showed that if

, then

. If we had expressed this function in the form

, we could have expressed the derivative as

. We could have conveyed the same information by writing

. Thus, for the function

, each of the following notations represents the derivative of

In place of

we may also use

Use of the

notation (called Leibniz notation) is quite common in engineering and physics. To understand this notation better, recall that the derivative of a function at a point is the limit of the slopes of secant lines as the secant lines approach the tangent line. The slopes of these secant lines are often expressed in the form

where

is the difference in the values corresponding to the difference in the values, which are expressed as

((Figure)). Thus the derivative, which can be thought of as the instantaneous rate of change of with respect to , is expressed as

Figure 1. The derivative is expressed as

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

gives the rate of change of a function

(or slope of the tangent line to

In (Figure) we found that for

. If we graph these functions on the same axes, as in (Figure), we can use the graphs to understand the relationship between these two functions. First, we notice that

is increasing over its entire domain, which means that the slopes of its tangent lines at all points are positive. Consequently, we expect

for all values of in its domain. Furthermore, as increases, the slopes of the tangent lines to

are decreasing and we expect to see a corresponding decrease in

. We also observe that

is undefined and that

, corresponding to a vertical tangent to

at 0.

Figure 2. The derivative

is positive everywhere because the function

is increasing.

In (Figure) we found that for

. The graphs of these functions are shown in (Figure). Observe that

is decreasing for

. For these same values of

. For values of

is increasing and

. Also,

has a horizontal tangent at

and

Figure 3. The derivative

where the function

is decreasing and

where

is increasing. The derivative is zero where the function has a horizontal tangent.

Use the following graph of

to sketch a graph of

The solution is shown in the following graph. Observe that

is increasing and

. Also,

is decreasing and

and on

. Also note that

has horizontal tangents at -2 and 3, and

and

Sketch the graph of

. On what interval is the graph of

above the -axis?

Now that we can graph a derivative, let’s examine the behavior of the graphs. First, we consider the relationship between differentiability and continuity. We will see that if a function is differentiable at a point, it must be continuous there; however, a function that is continuous at a point need not be differentiable at that point. In fact, a function may be continuous at a point and fail to be differentiable at the point for one of several reasons.

Let

be a function and be in its domain. If

is differentiable at , then

is continuous at .

is differentiable at , then

exists and

We want to show that

is continuous at by showing that

. Thus,

Therefore, since

is defined and

, we conclude that

is continuous at .

We have just proven that differentiability implies continuity, but now we consider whether continuity implies differentiability. To determine an answer to this question, we examine the function

. This function is continuous everywhere; however,

is undefined. This observation leads us to believe that continuity does not imply differentiability. Let’s explore further. For

This limit does not exist because

See (Figure).

Figure 4. The function

is continuous at 0 but is not differentiable at 0.

Let’s consider some additional situations in which a continuous function fails to be differentiable. Consider the function

Thus

does not exist. A quick look at the graph of

clarifies the situation. The function has a vertical tangent line at 0 ((Figure)).

Figure 5. The function

has a vertical tangent at

. It is continuous at 0 but is not differentiable at 0.

The function

also has a derivative that exhibits interesting behavior at 0. We see that

This limit does not exist, essentially because the slopes of the secant lines continuously change direction as they approach zero ((Figure)).

Figure 6. The function

is not differentiable at 0.

In summary:

We observe that if a function is not continuous, it cannot be differentiable, since every differentiable function must be continuous. However, if a function is continuous, it may still fail to be differentiable.
We saw that
failed to be differentiable at 0 because the limit of the slopes of the tangent lines on the left and right were not the same. Visually, this resulted in a sharp corner on the graph of the function at 0. From this we conclude that in order to be differentiable at a point, a function must be “smooth” at that point.
As we saw in the example of
, a function fails to be differentiable at a point where there is a vertical tangent line.
As we saw with
a function may fail to be differentiable at a point in more complicated ways as well.

A toy company wants to design a track for a toy car that starts out along a parabolic curve and then converts to a straight line ((Figure)). The function that describes the track is to have the form

, where and

are in inches. For the car to move smoothly along the track, the function

must be both continuous and differentiable at -10. Find values of and that make

both continuous and differentiable.

Figure 7. For the car to move smoothly along the track, the function must be both continuous and differentiable.

For the function to be continuous at

. Thus, since

and

, we must have

. Equivalently, we have

For the function to be differentiable at -10,

must exist. Since

is defined using different rules on the right and the left, we must evaluate this limit from the right and the left and then set them equal to each other:

We also have

This gives us

. Thus

and

Find values of and that make

both continuous and differentiable at 3.

and

The derivative of a function is itself a function, so we can find the derivative of a derivative. For example, the derivative of a position function is the rate of change of position, or velocity. The derivative of velocity is the rate of change of velocity, which is acceleration. The new function obtained by differentiating the derivative is called the second derivative. Furthermore, we can continue to take derivatives to obtain the third derivative, fourth derivative, and so on. Collectively, these are referred to as higher-order derivatives. The notation for the higher-order derivatives of

can be expressed in any of the following forms:

It is interesting to note that the notation for

may be viewed as an attempt to express

more compactly. Analogously,

For

, find

First find

Next, find

by taking the derivative of

Find

for

The position of a particle along a coordinate axis at time (in seconds) is given by

(in meters). Find the function that describes its acceleration at time .

Since

and

, we begin by finding the derivative of

Next,

Thus,

For

, find

For the following exercises, use the definition of a derivative to find

10.

For the following exercises, use the graph of

to sketch the graph of its derivative

11.

12.

13.

14.

For the following exercises, the given limit represents the derivative of a function

at . Find

and .

15.

16.

17.

18.

19.

20.

For the following functions,

sketch the graph and
use the definition of a derivative to show that the function is not differentiable at
.

21.

22.

23.

24.

For the following graphs,

determine for which values of the
exists but
is not continuous at , and
determine for which values of the function is continuous but not differentiable at .

25.

26.

, b.

27. Use the graph to evaluate a.

, b.

, c.

, d.

, and e.

, if they exist.

For the following functions, use

to find

28.

29.

30.

For the following exercises, use a calculator to graph

. Determine the function

, then use a calculator to graph

31. [T]

32. [T]

33. [T]

34. [T]

35. [T]

36. [T]

For the following exercises, describe what the two expressions represent in terms of each of the given situations. Be sure to include units.

37.

denotes the population of a city at time in years.

38.

denotes the total amount of money (in thousands of dollars) spent on concessions by customers at an amusement park.

a. Average rate at which customers spent on concessions in thousands per customer.
b. Rate (in thousands per customer) at which customers spent money on concessions in thousands per customer.

39.

denotes the total cost (in thousands of dollars) of manufacturing clock radios.

40.

denotes the grade (in percentage points) received on a test, given hours of studying.

a. Average grade received on the test with an average study time between two amounts.
b. Rate (in percentage points per hour) at which the grade on the test increased or decreased for a given average study time of hours.

41.

denotes the cost (in dollars) of a sociology textbook at university bookstores in the United States in years since 1990.

42.

denotes atmospheric pressure at an altitude of feet.

a. Average change of atmospheric pressure between two different altitudes.
b. Rate (torr per foot) at which atmospheric pressure is increasing or decreasing at feet.

43. Sketch the graph of a function

with all of the following properties:

for
for
and
and
does not exist.

44. Suppose temperature

in degrees Fahrenheit at a height in feet above the ground is given by

Give a physical interpretation, with units, of
.
If we know that
explain the physical meaning.

a. The rate (in degrees per foot) at which temperature is increasing or decreasing for a given height .
b. The rate of change of temperature as altitude changes at 1000 feet is -0.1 degrees per foot.

45. Suppose the total profit of a company is

thousand dollars when units of an item are sold.

What does
for
measure, and what are the units?
What does
measure, and what are the units?
Suppose that
. What is the approximate change in profit if the number of items sold increases from 30 to 31?

46. The graph in the following figure models the number of people

who have come down with the flu weeks after its initial outbreak in a town with a population of 50,000 citizens.

Describe what
represents and how it behaves as increases.
What does the derivative tell us about how this town is affected by the flu outbreak?

a. The rate at which the number of people who have come down with the flu is changing weeks after the initial outbreak.
b. The rate is increasing sharply up to the third week, at which point it slows down and then becomes constant.

For the following exercises, use the following table, which shows the height

of the Saturn V rocket for the Apollo 11 mission seconds after launch.