Derivative of inverse function

I'm reading the proof for the following theorem.

If f is a one-to-one differentiable function with inverse function and , then the inverse function is differentiable at *a* and

The Calculus textbook that I'm reading gives two proofs, the first one uses Newton's difference quotient, which I can understand.

The 2nd goes like this:

Replacing *a* by the general number *x* in the theorem, we get

If we write , then , so the equation can be expressed in Leibniz notation.

IF IT IS KNOWN IN ADVANCE THAT IS DIFFERENTIABLE, then its derivative can be computed more easily by using implicit differentiation.

My question is this, I don't understand which step of the proof requires the fact that is differentiable (Wondering)

Thanks a lot if you can help :)