Calculus- Early Transcendentals, 2021a

4.3. Derivative Rules 131
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It turns out that the Power Rule holds for any real number n (though it is a bit more difficult to prove).
Theorem 4.17: The Power Rule (General)
If n is any real number, then d dx (xn )=nx n−1 .
Example 4.18: Derivative of a Power Function
By the power rule, the derivative of g(x)=x 4 is g ′ (x)=4x 3 .
Theorem 4.19: The Constant Multiple Rule
If c is a constant and f is a differentiable function, then
d
dx [cf(x)] = c d dx f (x).
Proof. For convenience let g(x)=cf(x). Then:
g ′ g(x + h) − g(x)
(x)=lim
h→0 h
[ ]
f (x + h) − f (x)
= lim c
h→0 h
where c can be moved in front of the limit by the Limit Rules.
= lim
h→0
cf(x + h) − cf(x)
h
f (x + h) − f (x)
= c lim
= cf ′ (x),
h→0 h
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Example 4.20: Derivative of a Multiple of a Function
By the constant multiple rule and the previous example, the derivative of F(x)=2 · (17 + x 4 ) is
F ′ (x)=2(4x 3 )=8x 3 .
Theorem 4.21: The Sum/Difference Rule
If f and g are both differentiable functions, then
d
dx [ f (x) ± g(x)] = d dx f (x) ± d dx g(x).