# The rules to show a function is differentiable / continuously differentiable?

What are the main 'rules' or methods to show a function $f:\mathbb{R}^n \to \mathbb{R}^m$ is differentiable or continuously differentiable.

I know that to check differentiability at a particular point we have to check that all partial derivatives exist and are continuous at that point.

But what about showing a function is continuously differentiable? Does one just have to show that it's differentiable using the above and then differentiate it and show it's continuous? Is there perhaps a more direct proof that I've missed?

Ex: Check that $f(x,y)=x^2y^3$ is differentiable in $(0,0)$.
Solution: If you take partial derivatives you obtain $f_x(x,y)=2xy^3$ which is continuous in $(0,0)$, and $f_y(x,y)=3x^2y^2$ which is continuous in $(0,0)$. Therefore $f$ is differentiable in $(0,0)$.