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

176 Chapter 3 Differentiation Rules
Example 4
(a)
(b)
d
dx s3x 4 d − 3 d dx sx 4 d − 3s4x 3 d − 12x 3
d
dx s2xd − d dx fs21dxg − s21d d sxd − 21s1d − 21
dx ■
The next rule tells us that the derivative of a sum of functions is the sum of the
derivatives.
Using prime notation, we can write the
Sum Rule as
s f 1 td9 − f 9 1 t9
The Sum Rule If f and t are both differentiable, then
d
dx f f sxd 1 tsxdg − d dx f sxd 1 d dx tsxd
ProoF Let Fsxd − f sxd 1 tsxd. Then
F9sxd − lim
h l 0
Fsx 1 hd 2 Fsxd
h
− lim
h l 0
f f sx 1 hd 1 tsx 1 hdg 2 f f sxd 1 tsxdg
h
− lim
h l 0F
− lim
h l 0
f sx 1 hd 2 f sxd
h
f sx 1 hd 2 f sxd
h
1
tsx 1 hd 2 tsxd
h
G
1 lim
hl 0
tsx 1 hd 2 tsxd
h
(by Limit Law 1)
− f 9sxd 1 t9sxd
■
The Sum Rule can be extended to the sum of any number of functions. For instance,
using this theorem twice, we get
s f 1 t 1 hd9 − fs f 1 td 1 hg9 − s f 1 td9 1 h9 − f 9 1 t9 1 h9
By writing f 2 t as f 1 s21dt and applying the Sum Rule and the Constant Multiple
Rule, we get the following formula.
The Difference Rule If f and t are both differentiable, then
d
dx f f sxd 2 tsxdg − d dx f sxd 2 d dx tsxd
The Constant Multiple Rule, the Sum Rule, and the Difference Rule can be combined
with the Power Rule to differentiate any polynomial, as the following examples
demonstrate.
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