This is theorem 24.1 from the book "Analysis on Manifolds" by James Munkres:
Here is the beginning of the proof: 
I understand that if we show that $\alpha_{1}^{-1}$ is $C^{r}$ (which we can do by showing that it is locally $C^{r}$) then we show $\alpha_{1}^{-1} \circ \alpha_{0}$ is $C^{r}$. We show that $\alpha_{0}^{-1} \circ \alpha_{1}$ is $C^{r}$ using the same argument. The part that confuses me is the claim that the chain rule implies that the derivative of these functions is non-singular. Can someone please break it down for me?
Also, saying that $f = \alpha_{1}^{-1} \circ \alpha_{0}$ is $C^{r}$ means that it can be extended to a $C^{r}$ function on an open set. Is the claim that the derivative of $f$ is non-singular mean that the the derivative of the extended function is non-singular, or only on the domain of the function f before it was it was extended?