$C^{r-1}$ Diffeomorphisms on $C^r$ manifolds

Question

I have two questions about diffeomorphisms on manifolds and their tangent spaces.

1. Let $M$ and $N$ be $C^r$ manifolds, $r \geq 1$, and $f:M \rightarrow N$ be a $C^r$ diffeomorphism. I want to prove that $Tf:TM\rightarrow TN$ is a $C^{r-1}$ diffeomorphism.
2. Let $M$ and $N$ be $C^r$ manifolds, $r \geq 1$. I want to show that there is a $C^{r-1}$ diffeomorphism between $T(M\times N)$ and $T(M)\times T(N)$.

Actually, there is an exercise in Smooth Manifolds book (John M. Lee), that says if $M$ and $N$ are two smooth manifolds, then $T(M\times N)$ is diffeomorphic to $T(M)\times T(N)$, and I think I can use this to prove the second one, but I don't have any idea have can I do that!

Answer 1

I saw this exercise in Lee's book too. Here is a sketch, although I admit there are quite a few things to check, which I have not done (yet).

Fix $x_0\in X;\ y_0\in Y$. let $\pi_X,\ \pi_Y$ be the usual projections from the product. Finally, define $i_X,\ i_Y$ as follows: $i_X(x)=x\times y_0;\ i_Y(y)=x_0\times y$.

Of course, we have $d\pi_X:T{_{x_0\times y_0}}(X\times Y)\to T_{x_0}X: d\pi_X(v)f=v(f\circ\pi_X)$ and similarly for $d\pi_Y;\ di_X:T_{x_0}X\to T{_{x_0\times y_0}}(X\times Y): di_X(v)f=v(f\circ i_X)$ and similarly for $di_Y$.

Now, define $\Phi : T_{x_0} X \times T_{y_0} Y\to T_{(x_0,y_0)}(X \times Y)$ by $\Phi(v,w)(f)=di_X (v)f+di_Y (w)f$ and $\Psi(v)=(d\pi_X(v),d\pi_Y(v))$.

Of course, I need to show that $\Psi$ and $\Phi$ are diffeomorphic inverses.