Basic Analysis I — Real Analysis · Problem 4.1.8

The Derivative — Jiri Lebl

Mathematics — Real Analysis · derivative-chain-rule-power

Problem

Let

f : I \to \mathbb{R}

be differentiable. For

n \in \mathbb{Z}

, let

f^n

be the function defined by

f^n(x) \mathrel{\mathop:}= \bigl(f(x)\bigr)^n

. If

n < 0

, assume

f(x) \neq 0

for all

x \in I

. Prove that

(f^n)'(x) = n\bigl(f(x)\bigr)^{n-1} f'(x)