Example 4: Absolute Value Function

Case 1

Find the derivative of f(x) = |x|. x > 0. In this case |x| = x, and we have

Case 2

x < 0. Now |x| = - x, and

Case 3

x = 0. Then

and

The standard part of Δy/Δx is then 1 for some values of Δx and -1 for others. Therefore f'(x) does not exist when x = 0.

In summary,

f'(x)=	1	if x > 0,
	-1	if x < 0,
	undefined	if x = 0.

Figure 2.1.6 shows f(x) and f'(x).

The derivative has a variety of applications to the physical, life, and social sciences. It may come up in one of the following contexts.

• Velocity: If an object moves according to the equation s = f(t) where t is time and s is distance, the derivative v = f'(t) is called the velocity of the object at time t.
• Growth rates: A population y (of people, bacteria, molecules, etc.) grows according to the equation y = f(t) where t is time. Then the derivative y' = f'(t) is the rate of growth of the population y at time t.
• Marginal values (economics): Suppose the total cost (or profit, etc.) of producing x items is y = f(x) dollars. Then the cost of making one additional item is approximately the derivative y' = f'(x) because y' is the change in y per unit change in x. This derivative is called the marginal cost.