How can I prove the usual chain rule, but in the context of smooth manifolds? I mean: let $f$ and $g$ be two differentiable maps (from $M$ to $N$ and from $N$ to $P$, respectively) and where all are smooth manifolds. Also, let $p$ be a point of $M$. I want to show that the usual chain rule: $d(g \circ f)(p) = (dg)(f(p)) \circ (df)(p)$, where $d$ denotes the differential, still applies. Even if its really easy, or straight from the definitions. I'm still getting used to differential geometry notation and I have only problems and no solutions or examples to "practice the language". Thank you a lot for the patience.