Here is a part of a lecture note:
I need some help to solve the exercise.
I want to show that if $\psi\circ f\circ\phi^{-1}$ is differentiable and $\alpha, \psi$ and $\beta,\phi$ are in the same charts, i.e, if $\psi^{-1}\circ\alpha$ and $\beta^{-1}\circ \phi$ are differentiable, then $\alpha\circ f\circ\beta^{-1}$ is differentiable. As $\phi^{-1}\circ \phi$ and $\psi^{-1}\circ\psi$ are differentiable, so is $\psi^{-1}\circ\psi\circ f \circ \phi^{-1}\circ \phi=f$. How do I insert $\alpha$ and $\beta$? Can I say that $\psi\circ\psi^{-1}\circ\psi^{-1}\circ\alpha$ and similarly $\beta^{-1}\circ\psi\circ\psi\circ\psi^{-1}$ are differentiable which means that their composition with $f$ is differentiable?