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
A function
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
In place of
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
In (Figure) we found that for
In (Figure) we found that for
Use the following graph of
The solution is shown in the following graph. Observe that
Sketch the graph of
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
If
We want to show that
Therefore, since
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 limit does not exist because
See (Figure).
Let’s consider some additional situations in which a continuous function fails to be differentiable. Consider the function
Thus
The function
This limit does not exist, essentially because the slopes of the secant lines continuously change direction as they approach zero ((Figure)).
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
For the function to be continuous at
and
For the function to be differentiable at -10,
must exist. Since
We also have
This gives us
Find values of and that make
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
It is interesting to note that the notation for
For
First find
Next, find
Find
The position of a particle along a coordinate axis at time (in seconds) is given by
Since
Next,
Thus,
For
For the following exercises, use the definition of a derivative to find
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
For the following exercises, use the graph of
11.
12.
13.
14.
For the following exercises, the given limit represents the derivative of a function
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.
a.
b.
23.
24.
a.
b.
For the following graphs,
- determine for which values of the exists butis not continuous at , and
- determine for which values of the function is continuous but not differentiable at .
25.
26.
a.
27. Use the graph to evaluate a.
For the following functions, use
28.
29.
30.
For the following exercises, 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.
38.
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.
40.
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.
42.
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
- for
- for
- and
- and
- does not exist.
44. Suppose temperature
- 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
- What does formeasure, 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
- 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
0 | 0 |
1 | 2 |
2 | 4 |
3 | 13 |
4 | 25 |
5 | 32 |
47. What is the physical meaning of
48. [T] Construct a table of values for
0 | 2 |
1 | 2 |
2 | 5.5 |
3 | 10.5 |
4 | 9.5 |
5 | 7 |
49. [T] The best linear fit to the data is given by
50. [T] The best quadratic fit to the data is given by
51. [T] The best cubic fit to the data is given by
52. Using the best linear, quadratic, and cubic fits to the data, determine what