What is a closed form expression for the ∂/∂w(∂t/∂w) if w(t) is complicated function? Lets say we have a trigonometric function w(t) that can not be inverted as t(w).
The derivative ∂t/∂w can be calculated as 1/(∂w(t)/∂t).
What is a closed form expression for the second derivative ∂/∂w(∂t/∂w) ?
Numerically do you start from the closed form or there is easier way ?
This represents the group delay dispersion and third order dispersion.

If I understand correctly you are interested in a closed form for the second derivative of the inverse function expressed in terms of other information. Then:
$f(f^{-1}(x))=x$
$f'(f^{-1}(x))\cdot(f^{-1})'(x)=1$ (differentiating the previous identity)
$f''(f^{-1}(x))\cdot((f^{-1})'(x))^{2}+f'(f^{-1}(x))(f^{-1})''(x)=0$ (differentiating again)
so $(f^{-1})''(x)=\frac{-f''(f^{-1}(x))\cdot((f^{-1})'(x))^{2}}{f'(f^{-1}(x))}=\frac{-f''(f^{-1}(x))}{(f'(f^{-1}(x)))^{3}}$
Note that you need not invert the function to compute this. If $f(a)=b$ and you want the derivative of $f^{-1}$ at $x=a$ then $f^{-1}(a)=b$ so the inverse drops from the closed form.
