How to prove a function $f :R^2 \rightarrow R^2$ is differentiable?

Question

I want to prove a function $f :R^2 \rightarrow R^2$ is differentiable in general. Is it sufficient to prove that the four partial derivatives exist at every point in $R^2$? In a text book I found the definition for differentiability.

$\lim_{h\rightarrow0} \frac{||f(x+h)-f(x)-T(h)||}{||h||} =0$ , where $T:R^2\rightarrow R^2, x \,\in R^2 $

But how does one find such a suitable $T(x)$?