If $E\subset \mathbb{R}$, $f:E\to\mathbb{R}^m$, and $||f(x)||=1$ for all $x\in E$, then $f'(x)\cdot f(x)=0$ for all $x$

Question

I'm given the hint that $||f(x)||=1$ can be written as $||f(x)||^2=1$, and that the chain rule may be helpful, but I'm still not sure how to proceed. Can someone point me in the right direction?

Answer 1

Hint: the differential of $\mid f(x)\mid^2=\langle x,x\rangle$ is zero