Let's suppose I have a smooth vector field $X$ on $\mathbb{R}^n$ and I have a integral curve of $X$: $$\gamma^x_m(t):t \in (-\epsilon,\epsilon)\rightarrow \mathbb{R}^n, \qquad \gamma^x_m (0)=m.$$
I want to calculate $\frac{d}{dt}_{|t=0}X(\gamma^x_m(t))$.
Intuitively I would use the "chain rule" and do something like this: $\frac{d}{dt}_{|t=0}X(\gamma^x_m(t))=\frac{d}{d \gamma^x_m} X(\gamma^x_m(t))\frac{d}{dt}\gamma^x_m(t)_{|t=0}$.
How can I justify this? I've browsed through my literature and can't find anything useful.
Is $\frac{d}{d \gamma^x_m} X(\gamma^x_m(t))$ maybe equal to the pushforward of $X$ given by a Jacobian, since we have $X$ on $\mathbb{R}^n$ ?
How can my "chain rule" be justified when $X$ is defined on some (general) smooth manifold, instead of $\mathbb{R}^n$ ?