5128_Ch03_pp098-184

116 Chapter 3 Derivatives
3.3
What you’ll learn about
• Positive Integer Powers, Multiples,
Sums, and Differences
• Products and Quotients
• Negative Integer Powers of x
• Second and Higher Order
Derivatives
. . . and why
These rules help us find
derivatives of functions
analytically more efficiently.
Rules for Differentiation
Positive Integer Powers, Multiples,
Sums, and Differences
The first rule of differentiation is that the derivative of every constant function is the zero
function.
RULE 1
Derivative of a Constant Function
If f is the function with the constant value c, then
df
d
c) 0.
d x d x
Proof of Rule 1 If f x c is a function with a constant value c, then
lim f x h f x
lim c c
lim 0 0.
h→0 h
h→0 h h→0
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The next rule is a first step toward a rule for differentiating any polynomial.
RULE 2
Power Rule for Positive Integer Powers of x
If n is a positive integer, then
d
x d x
n nx n1 .
Proof of Rule 2 If f x x n , then f x h x h n and the difference quotient
for f is
x h n x
.
n
h
We can readily find the limit of this quotient as h→0 if we apply the algebraic identity
a n b n a ba n1 a n2 b … ab n2 b n1 n a positive integer
with a x h and b x. For then a b h and the h’s in the numerator and denominator
of the quotient cancel, giving
f x h f x
x h n x
n
h
h
hx h n1 x h n2 x … x hx n2 x n1
h
x h n1 x h n2 x … x hx n2 x n1 .
n terms, each with limit x n1 as h→0
Hence,
d
x d x
n lim f x h f x
nx
h→0 h
n1 .
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