Let $M$ and $N$ be smooth manifolds and $f: M\rightarrow N$ a diffeomorphism. Prove that the map $df:TM \rightarrow TN$ is a homeomorphism.

Question

I am going through qualifying exam questions and I am stuck on this problem. I don't think it should be too difficult, but I am having a lot of difficulty. I am not even sure how to start. Some hints would be appreciated. Thank you.

Let $M$ and $N$ be smooth manifolds and $f: M\rightarrow N$ a diffeomorphism. Prove that the map $df:TM \rightarrow TN$ is a homeomorphism.

Answer 1

To expand a little on my hint in the comments: you only have to prove that $d(fg) = df dg$ and that $d(1_M) = 1_{TM}$, then you should be able to figure out how a diffeomorphism induces an invertible linear map of tangent spaces.