Proof of theorem on continous differentiability of the inverse function

Question

How can I prove the following statement?

Consider a function $p:[c,d] \to [a,b]$ such that

1. $p \in C^1([a,b])$
2. $p$ is a bijection between $[a,b]$ and $[c,d]$
3. $p'(r)\neq0$ $\,\,\,\,\forall r \in [a,b]$

Then the inverse is $C^1$ in $[c,d]$, that is $$p^{-1}:[a,b] \to [c,d] \,\,\,\,\, , \,\, \,\,\, p^{-1} \in C^1([c,d])$$

I would like to know in particular if all the 1.,2. and 3. conditions are used and how.

(If the notation $C^k$ is not clear see e.g. here)

Answer 1

HINT

Use the derivative of the inverse function formula