$G$ a compact Lie group, $W$ a smooth G-manifold, $x \in W$.
Then as $G$ is compact $G \cdot x$ is an embedded submanifold.
Now the text I'm reading claims that for $g \in G_x$,
$dg_x : \text{T}_x W \to \text{T}_{g\cdot{x}}W=\text{T}_x W$
sends the tangent space of the orbit to itself by the identity map. However, I'm having difficulty showing this is actually the identity map on $\text{T}_{x}G\cdot{x}$. I've tried passing to $G/G_x$ but as $G$ is not necessarily Abelian I don't think this is any easier. Is there something obvious I'm missing?