Basic Analysis I — Real Analysis · Problem 4.4.4

The Derivative — Jiri Lebl

Mathematics — Real Analysis · odd-nth-root

Problem

Let

n \in \mathbb{N}

be odd and

n \geq 3

. Prove that every

x

has a unique

n

th root. That is, there exists a number

y

such that

y^n = x

. Prove that the function defined by

g(x) \mathrel{\mathop:}= y

is differentiable except at

x = 0

and compute the derivative. Prove that

g

is not differentiable at

x = 0