Math Help - tangent spaces

1. tangent spaces

It is an easy problem, but I am not sure that my argument is correct.

Suppose that f: X ­-> Y is a diffeomorphism, and prove that at each x
its derivative dfx is an isomorphism of tangent spaces.

2. Let g:Y -> X be the inverse. Then d(f o g)(y) = d(1_Y)(y)=1_{Ty} and similarily d(g o f)(x) = d(1_X)(x)=1_{Tx}. Using the chain rule, you have your result!

Next time, you can post your solution and we can look at what you did.