Derivative of a homogeneous function of degree n

Question

How do we show that, if $f:\mathbb{R}^n \rightarrow \mathbb{R}^m$ is homogeneous of degree $d > 1$, and if $f$ is differentiable at $0$, then $f'(0)$ is the zero map from $\mathbb{R}^n$ to $\mathbb{R}^m$?

Answer 1

For $|x|\to 0$, $$ f(x) = |x|^d f\left(\frac{x}{|x|} \right)= O(|x|^d) $$ because if $f$ is continuous it is bounded on the unit ball.

In particular, as $d>1$ $f$ is locally linear around 0 with derivative $f'(0) = 0$.