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Lebl-Basic-Analysis-I
· Problem 4.1.8
The Derivative
Mathematics · derivative-chain-rule-power
Problem
Let
f
:
I
→
R
f : I \to \mathbb{R}
f
:
I
→
R
be differentiable. For
n
∈
Z
n \in \mathbb{Z}
n
∈
Z
, let
f
n
f^n
f
n
be the function defined by
f
n
(
x
)
:
=
(
f
(
x
)
)
n
f^n(x) := \bigl(f(x)\bigr)^n
f
n
(
x
)
:=
(
f
(
x
)
)
n
. If
n
<
0
n < 0
n
<
0
, assume
f
(
x
)
≠
0
f(x) \neq 0
f
(
x
)
=
0
for all
x
∈
I
x \in I
x
∈
I
. Prove that
(
f
n
)
′
(
x
)
=
n
(
f
(
x
)
)
n
−
1
f
′
(
x
)
(f^n)'(x) = n\bigl(f(x)\bigr)^{n-1} f'(x)
(
f
n
)
′
(
x
)
=
n
(
f
(
x
)
)
n
−
1
f
′
(
x
)
.
Related theory
Power In Ac Circuits And Power Factor
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