As the title says I have to show that $ h \in C^n(\mathbb{R}^2)$, but I do not know how to show it.
I know that the partial derivatives of h have to exist and be continous in order for h to be $C^n$, but how do I show this?
Do I start out with $n=1$ and then move on until I see a pattern?
Here is a hint. Suppose that $f$ and $g$ are two functions defined on $\mathbb{R}$; define $h \colon \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ by $h(u,v)=f(u)g(v)$. Convince yourself that if $f$ and $g$ are differentiable, then so is $h$. Now iterate $n$ times.
To apply this to your case, remark that your map can be decomposed into the sum of three functions of the previous form.