Definition:Differentiable Mapping between Manifolds/Point/Definition 1

Definition
$f$ is differentiable at $p$ for every pair of charts $(U, \phi)$ and $(V,\psi)$ of $M$ and $N$ with $p\in U$ and $f(p)\in V$:
 * $\psi\circ f\circ \phi^{-1} : \phi ( U \cap f^{-1}(V)) \to \psi(V)$

is differentiable at $\phi(p)$.