Definition:Differentiable Mapping between Manifolds/Point/Definition 1

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

is differentiable at $\phi \left({p}\right)$.