Let $G$ be a Lie group acting on a manifold $M$. Fix a point $m \in M$ and we denote by $a_m$ the map given by $$a_m : G \rightarrow M, g \rightarrow g.m $$ the action of of $G$ on $m$.
I read that we have that $T_m O_m = Im({(a_m)}_*)$, where $T_m O_m$ is the tangent space of the orbit $O_m$ and ${(a_m)}_*$ denotes the differential at the identity of the map $a_m$.
It easy to see that $Im({(a_m)}_*) \subset T_m O_m$, but I don't know how to prove the other inclusion ?