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.

Aloizio's answer should already be enough (and you should accept it), but explicitly, we'd have $$p(x_1,\cdots,x_n) = \sum a_{i_1\cdots i_n} x_1^{i_1}\cdots x_n^{i_n}.$$Since $$\Bbb R^n \ni (x_1,\cdots,x_n) \mapsto x_i \in \Bbb R$$is differentiable for each $i$, it follows that $p$ is differentiable as well.
