Use the chain rule to show that $||x||^4/1+||x||^2$ is differentiable from $\mathbb{R}^n \to \mathbb{R}$?

Question

I know I want two differentiable functions but I'm having trouble coming up with two functions and showing that they are differentiable from $\mathbb{R}^n \to \mathbb{R}$.

My attempt was to use $g=x^4/1+x$ and $h=|x|$ but that doesn't work.

Answer 1

The functions are $h(x)=\|x\|^2$, and $g(t)=\frac{t^2}{1+t}$. As $h(x)\geq0$ for all $x$, the domain we need for $g$ is $t\geq0$, and $f$ is differentiable there. So $$ f(x)=g(h(x)), $$ with both $g$ and $h$ differentiable.