I do not really like the notation,

and .

It makes me want to vomit.

Let us assume,

has an inverse on some open interval

is the inverse function.

Further, assume is twice differenciable on the interval.

Then, is twice differenciable on the interval.

Thus,

throughout the interval.

Take derivative (chain rule),

Thus,

*)

Take derivative again, use chain on left use quotient on right.

Thus, (I assume, but am lazy to check that).

Thus,

In your notation,

*)Note at some point. There are two possibilities. Either zero throught the interval in that case, but then cannot exists, because it is one-to-one map. And it cannot happen that it is zero at some point but not all, that will lead to non-differenciability. Thus, we can divide.