In a few physics papers (lattice gauge theory papers, to be more specific) I've seen the following definition for differentiation on group space $$ \frac{\partial}{\partial U} f(U) = \frac{\partial}{\partial \alpha_a} f(e^{i \alpha_a T^a} U)\,, $$ where $U \in SU(N)$ and the $T^a$ are the generators of the group. My questions are: why is it defined in such a way? Mathematically speaking, how can it be proved?
Thanks in advance.