Let $f\colon M \rightarrow N$ be a smooth map. If $f$ is a diffeomorphism I am trying to show that the linear map $f_*$ : $T_pM \rightarrow T_{f(p)}M$ is an isomorphism for all $p \in M$. I know the derivative map $T_pf\colon T_pM \rightarrow T_{f(p)}N$ is an isomorphism, but I have trouble to proving when the map defines $T_pM \rightarrow T_{f(p)}M$.
Thank you for your help.