Polynomial n variables differentiable on $\mathbb{R^n}$ How would I define a polynomial of $n$ variables? And how would I go on to prove that any polynomial in $n$ variables is differentiable on $\mathbb{R^n}$? (assuming this function is continuous on $\mathbb{R^n}$)
I'm struggling to find a formal definition and I assume I use the chain rule for the second part but other than that I'm not sure what to do.

You can think of a polynomial function (*) in $\mathbb{R}^n$ as a composition of multiplications and sums of the projections $\pi_i$ and constant maps (**).
That this is differentiable follows from differentiability of the projections and sums/products of differentiable functions.
(*) Sidenote - a polynomial of $n$ variables is not, in general, a polynomial function $f:\mathbb{R}^n \to \mathbb{R}$. It has a more "formal" character.
(**) Sidenote 2 - To give an explicit definition:
Define a finite R-multi-index $I$ to be an element of $\mathbb{R} \times \Bbb{N}^n$, and $M^I: \mathbb{R}^n \to \mathbb{R} $ to be $M^I:=c \pi_1^{j_1} \cdots \pi_n ^{j_n},$ where $I=(c,j_1,\cdots, j_n)$ is a finite R-multi-index. Now, $f:\Bbb{R}^n \to \Bbb{R}$ is a polynomial function  if there exists a finite amount $N$ of finite R-multi-indices $I_i$ such that
$$f=\sum_{i=1}^N M^{I_i}.$$
(Note that this is equivalent to Ivo's formula, with the slight difference that we have a somewhat more explicit control over the summation index).
