Page 58 - 2024-calc4e-SE proofs-4e.indd

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

208 Chapter 2 • The Derivative and Its Properties

NOTE In Section 2.3, we proved the Simple
d THEOREM Simple Power Rule
n
Power Rule, x = nx n − 1 where n is a
n
dx The derivative of y = x , where n is any integer, is
positive integer. Here we have extended the
Simple Power Rule from positive integers to
all integers. In Chapter 3, we extend the ′ d n n − 1
y = x = nx
result to include all real numbers. dx
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EXAMPLE 5 Differentiating Using the Simple Power Rule

d −1 −2 1
(a) x = −x = −
dx x 2
d 1 d −2 −3 2
(b) = u = −2u = −
du u 2 du u 3
d 4 d −5 −6 −6 20
(c) = 4 s = 4 · (−5) s = −20s = −
ds s 5 ds s 6

R
NOW WORK Problem 31 and AP Practice Problem 5.

EXAMPLE 6 Using the Simple Power Rule in Electrical Engineering
Ohm’s Law states that the current I running through a wire is inversely proportional
V
to the resistance R in the wire and can be written as I = , where V is the voltage.
R
Find the rate of change of I with respect to R when V = 12 volts.

Solution
dI
The rate of change of I with respect to R is the derivative . We write Ohm’s Law
d R
V
with V = 12 as I = = 12R −1 and use the Simple Power Rule.
R

dI d −1 d −1 −2 12
= (12R ) = 12 · R = 12(−1R ) = −
d R d R d R R 2
dI
The minus sign in indicates that the current I decreases as the resistance R in the
d R
wire increases.

NOW WORK Problem 91.

3 Find Higher-Order Derivatives

Since the derivative f is a function, it makes sense to ask about the derivative of f .
′
′
′
The derivative (if there is one) of f is also a function called the second derivative of f
and denoted by f , read “ f double prime.”
′′
By continuing in this fashion, we can find the third derivative of f , the fourth
derivative of f, and so on, provided that these derivatives exist. Collectively, these are
called higher-order derivatives.

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