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

x
Section 2.3 • The Derivative of a Polynomial Function; The Derivative of y = e and y = ln x 193

THEOREM Derivative of f (x) = x
If f (x) = x, then

d
′
f (x) = x = 1
dx
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y 2
When n = 2, then f (x) = x is the square function. The derivative of f is
8
2
2
2
6 d (x + h) − x 2 x + 2hx + h − x 2
2
(!2 , 4) f (x) = x = lim = lim
′
4 h→0 h→0
f(!2) !4 dx h h
(1 , 1)
2 h(2x + h)
f(1) 2 = lim = lim(2x + h) = 2x
h→0 h h→0
!2 2 4 x
f(0) 0 2
The slope of the tangent line to the graph of f (x) = x is different for every
Figure 25 f (x) = x 2 number x. Figure 25 shows the graph of f and several of its tangent lines. Notice that
the slope of each tangent line drawn is twice the value of x.
3
When n = 3, then f (x) = x is the cube function. The derivative of f is
2
3
3
2
3
(x + h) − x 3 x + 3x h + 3xh + h − x 3
f (x) = lim = lim
′
h→0 h h→0 h
2
2
h(3x + 3xh + h ) 2 2 2
= lim = lim(3x + 3xh + h ) = 3x
h→0 h h→0
Notice that the derivative of each of these power functions is another power
function, whose degree is 1 less than the degree of the original function and whose
coefficient is the degree of the original function. This rule holds for all power functions
as the following theorem, called the Simple Power Rule, indicates.
THEOREM Simple Power Rule
n
The derivative of the power function y = x , where n ≥ 1 is an integer, is
IN WORDS The derivative of x raised to an
integer power n ≥ 1 is n times x raised to the
d
power n − 1. ′ n n − 1
y = x = nx
dx
n
n
NEED TO REVIEW? The Binomial Theorem is Proof If f (x) = x and n is a positive integer, then f (x + h) = (x + h) . We use the
n
discussed in Section P.8, pp. 76–77. Binomial Theorem to expand (x + h) . Then
n(n − 1) n − 2 2 n(n − 1)(n − 2) n − 3 3 n − 1 n
n
n
n − 1
f (x + h) = (x + h) = x + nx h + x h + x h + · · · + nxh + h
2 6
f (x + h) − f (x)
f (x) = lim
′
h→0 h
n (n − 1) n − 2 2 n(n − 1)(n − 2) n − 3 3 n − 1

n − 1
n
x + nx h + x h + x h + · · · + nxh + h n − x n
2 6
= lim
h→0 h
n(n − 1) n(n − 1)(n − 2)
h + · · · + nxh
nx n − 1 h + x n − 2 2 x n − 3 3 n − 1 + h n
h +
2 6
= lim Simplify.
h→0 h
(Proof continues on page 194.)
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