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

194 Chapter 2 • The Derivative and Its Properties

n(n − 1) n(n − 1)(n − 2)
h nx n − 1 + x n − 2 h + x n − 3 2 n − 2 + h n − 1
h + · · · + nxh
2 6
= lim Factor h in the numerator.
h→0 h
n(n − 1) n − 2 n(n − 1)(n − 2) n − 3 2 n − 2 n − 1

n − 1
= lim nx + x h + x h + · · · + nxh + h Divide out the common h.
h→0 2 6
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= nx n − 1 Take the limit. Only the first
term remains.
EXAMPLE 2 Differentiating a Power Function Using the Simple Power Rule

d d
5
n
9
10
NOTE x = nx n − 1 is true not only for (a) x = 5x 4 (b) If g(x) = x , then g (x) = 10x .
′
dx dx
positive integers n but also for any real
number n. But the proof requires future
results. As these are developed, we will NOW WORK Problem 1 and AP Practice Problem 1.
R
expand the Simple Power Rule to include
an ever-widening set of numbers until we
arrive at the fact it is true when n is a real But what if we want to find the derivative of the function f (x) = ax when a 6= 1?
n
number.
The next theorem, called the Constant Multiple Rule, provides a way.
THEOREM Constant Multiple Rule
If a function f is differentiable and k is a constant, then F(x) = k f (x) is a function
that is differentiable and
IN WORDS The derivative of a constant
′
′
times a differentiable function f equals the F (x) = k f (x)
constant times the derivative of f .
Proof Use the definition of a derivative, Form (2).
F(x + h) − F(x) k f (x + h) − k f (x)
F (x) = lim = lim
′
h→0 h h→0 h
k [ f (x + h) − f (x)] f (x + h) − f (x)
= lim = k · lim = k · f (x)
′
h→0 h h→0 h
Using Leibniz notation, the Constant Multiple Rule takes the form

d d
[k f (x)] = k f (x)
dx dx
A change in the symbol used for the independent variable does not affect the
d 2 d 5 4
derivative formula. For example, t = 2t and u = 5u .
dt du

EXAMPLE 3 Differentiating a Constant Times a Power Function
Find the derivative of each function:

1
4 3
(a) f (x) = 5x 3 (b) g(u) = − u 2 (c) u(x) = π x
2
Solution
Notice that each of these functions involves the product of a constant and a power
function. So, we use the Constant Multiple Rule followed by the Simple Power Rule.

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