Let $f$ be a diffeomorphism of $M$ onto $N$ (both $M$,$N$ are smooth manifold). Prove that the mapping $f_{*x}$ is a vector-space isomorphism between $T_xM$ and $T_xN$.
Here is a proof:
Let $x=f^{-1}(y) \in M,\, y\in N$ and $h\in M, k\in N$ and I denote $U=f_{*x}$ and $V=f^{-1}_{x*}$. We know that $U,V$ are linear maps.
We have: $$ f(x+h)=f(x)+Uh+o(|h|) $$ $$ f^{-1}(y+k)=f^{-1}(y)+Vk+o(|k|) $$
Then $$ y+k = f(f^{-1}(y)+Vk+o(|k|))=y+U(Vk+o(|k|)) +o(|Vk+o(|k|)|) $$ which yields $$ k=UVk $$ for every $k$ so $U$ is invertible and $U^{-1}=V$.
Denote by $g=f^{-1}$ we have using the chain rule,
$$ d(g\circ f)[x](h) = dg[f(x)](df[x](h)) $$ which gives $$ h = V(Uh) $$ hence $U^{-1}=V$.