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Sullivan 04 apcalc4e 45342 ch02 166 233 5pp August 7, 2023 12:54

Section 2.2 • The Derivative as a Function; Differentiability 179

2.2 The Derivative as a Function; Differentiability

OBJECTIVES When you finish this section, you should be able to:
1 Define the derivative function (p. 179)
2 Graph the derivative function (p. 181)
3 Identify where a function is not differentiable (p. 182)
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4 Explain the relationship between differentiability and continuity (p. 184)

1 Define the Derivative Function

The derivative of a function f at a real number c has been defined as the real number

f (x) − f (c)
Form (1) f (c) = lim
′
x→c x − c
provided the limit exists. We refer to this representation of the derivative as Form (1).
Another way to find the derivative of f at any real number is obtained by rewriting the
f (x) − f (c)
expression and letting x = c + h, h 6= 0. Then
x − c
f (x) − f (c) f (c + h) − f (c) f (c + h) − f (c)
y = =
x − c (c + h) − c h
Tangent (c 1 h, f (c 1 h))
line Since x = c + h, then as x approaches c, h approaches 0. Form (1) of the derivative
Secant line with these changes becomes
(c, f(c)) y 5 f(x) f (x) − f (c) f (c + h) − f (c)
f (c) = lim = lim
′
h f (c 1 h) 2 f (c) x→c x − c h→0 h
So now, we have an equivalent way to write Form (1) for the derivative of f at a real
c 0 ← h x 5 c 1 h x number c.
c ← x
f (c + h) − f (c)
f (c) = lim
′
Figure 7 The slope of the tangent line at h→0 h
f (c + h)− f (c)
c is f (c) = lim .
′
h→0 h See Figure 7.
′
In the expression above for f (c), c is any real number. That is, the derivative f ′
is a function, called the derivative function of f . Now replace c by x, the independent
variable of f .
DEFINITION The Derivative Function f ′
The derivative function f of a function f is
′
IN WORDS In Form (2) the derivative is the f (x + h) − f (x)
′
limit of a difference quotient. Form (2) f (x) = lim
h→0 h
provided the limit exists. If f has a derivative, then f is said to be differentiable.

We refer to this representation of the derivative as Form (2).

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