How to differentiate the function $f(\mathbf x) = \|\mathbf x\|^2 \mathbf x$? 
Let $f:\mathbb R^n\to\mathbb R^n$ be given by the equation $f(\mathbf x)=\|\mathbf x\|^2 \mathbf x$. Show that $f$ is of class $C^\infty$ and that $f$ carries the unit ball $B(\mathbf 0;1)$ onto itself in a one-to-one fashion. Show, however, that the inverse function is not differentiable at $\mathbf 0$. 

How does one differentiate a function involving the Euclidean norm? It's simple enough if it was just the norm itself, but multiplied by a vector I'm not sure how to go about it.

Hint: To make things easier for you, let's work on $n=2$ as always...
$f(x) = f(x_1,x_2) = \begin{pmatrix} (x_1^2+x_2^2)x_1 \\ (x_1^2+x_2^2)x_2 \end{pmatrix}$
