James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

144 Chapter 2 Limits and Derivatives
The velocity of the ball as it hits the ground is therefore
vSÎ
4.9D 450 − 9.8Î 450 < 94 mys n
4.9
Derivatives
We have seen that the same type of limit arises in finding the slope of a tangent line
(Equation 2) or the velocity of an object (Equation 3). In fact, limits of the form
lim
h l0
f sa 1 hd 2 f sad
h
arise whenever we calculate a rate of change in any of the sciences or engineering, such
as a rate of reaction in chemistry or a marginal cost in economics. Since this type of limit
occurs so widely, it is given a special name and notation.
f 9sad is read “ f prime of a.”
4 Definition The derivative of a function f at a number a, denoted by
f 9sad, is
if this limit exists.
f 9sad − lim
h l0
f sa 1 hd 2 f sad
h
If we write x − a 1 h, then we have h − x 2 a and h approaches 0 if and only if x
approaches a. Therefore an equivalent way of stating the definition of the derivative, as
we saw in finding tangent lines, is
5
f 9sad − lim
x l a
f sxd 2 f sad
x 2 a
ExamplE 4
Find the derivative of the function f sxd − x 2 2 8x 1 9 at the number a.
SOLUtion From Definition 4 we have
Definitions 4 and 5 are equivalent, so
we can use either one to compute the
derivative. In practice, Definition 4
often leads to simpler computations.
f 9sad − lim
h l0
f sa 1 hd 2 f sad
h
fsa 1 hd 2 2 8sa 1 hd 1 9g 2 fa 2 2 8a 1 9g
− lim
h l0
h
a 2 1 2ah 1 h 2 2 8a 2 8h 1 9 2 a 2 1 8a 2 9
− lim
h l0
h
2ah 1 h 2 2 8h
− lim
− lim s2a 1 h 2 8d
h l0 h
h l0
− 2a 2 8
n
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