It is easy to get a expression for the nth-derivative of an inverse function ; starting from $(f^{-1})'=\frac{1}{f'\circ f^{-1}}$, one gets things like $(f^{-1})^{(n)}=\frac{\sum a\_k\prod (f^{(n\_j)}\circ f^{-1})^j}{(f'\circ f^{-1})^{2n-1}}$, with reasonably easy constraints on the $n\_j$. But what are the values of the $a\_k$? I believe I read somewhere this was an application of umbral calculus, but I don't see how, and inverting Faà di Bruno's formula on the identity $f\circ f^{-1}=id$ don't seem to get anywhere.



Faà di Bruno's formula for inverse functions?
Riordan's **Combinatorial identities** has a chapter on partition polynomials that may be helpful. It specifically covers the question you are asking, but is in umbral calculus. 


