In Progress
Lesson 14 of 24
In Progress

# The Derivative as a Function

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.

# Derivative Functions

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.

### Finding the Derivative of a Square-Root Function

Find the derivative of .

### Finding the Derivative of a Quadratic Function

Find the derivative of the function .

#### Solution

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

Find the derivative of .

#### Solution

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  or . 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.

# 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  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  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.

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  has a horizontal tangent at  and .

### Sketching a Derivative Using a Function

Use the following graph of  to sketch a graph of .

#### Solution

The solution is shown in the following graph. Observe that  is increasing and . Also,  is decreasing and  on  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?

#### Hint

The graph of  is positive where  is increasing.

# Derivatives and Continuity

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.

### Differentiability Implies Continuity

Let  be a function and  be in its domain. If  is differentiable at , then  is continuous at .

## Proof

If  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.

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)).

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)).

In summary:

1. 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.
2. 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.
3. As we saw in the example of , a function fails to be differentiable at a point where there is a vertical tangent line.
4. As we saw with  a function may fail to be differentiable at a point in more complicated ways as well.

### A Piecewise Function that is Continuous and Differentiable

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.

#### Solution

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

# Higher-Order Derivatives

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, .

### Finding a Second Derivative

For , find .

#### Solution

First find .

Next, find  by taking the derivative of .

Find  for .

### Finding Acceleration

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 .

#### Solution

Since  and , we begin by finding the derivative of :

Next,

Thus, .

For , find .

### Key Concepts

• The derivative of a function  is the function whose value at  is .
• The graph of a derivative of a function  is related to the graph of . Where  has a tangent line with positive slope,  has a tangent line with negative slope, . Where  has a horizontal tangent line, .
• If a function is differentiable at a point, then it is continuous at that point. A function is not differentiable at a point if it is not continuous at the point, if it has a vertical tangent line at the point, or if the graph has a sharp corner or cusp.
• Higher-order derivatives are derivatives of derivatives, from the second derivative to the  derivative.

# Key Equations

• The derivative function