Show total differentiability of $phi: mathbb R^n to mathbb R^n, x mapsto varphi(lVert xrVert_2) x$ where $varphi$ is differentiable

Quick Proof

We know that

$varphi$ is differentiable. $$|cdot |:xto |x|$$ is differentiable everywhere except at zero.

By composition :

$$ phi=(varphi circ | cdot |) ×Id$$

is differentiable.

Definition proof

$f(x) triangleq D(| cdot |) (x)$ exits for all non null $x$

$phi(x+h) =varphi(|x+h|)(x+h) $

Yet

$varphi(|x+h|)=varphi(|x|+f(x)(h) +o(|h|)) =varphi(|x|) +Dvarphi(|x|)(f(x)(h) +o(h)) $

Can you finish from here?