The formula for higher order derivatives of compound functions is known as Faà di Bruno's formula. Does there exist a similar formula for higher order derivatives of an inverse function, i.e. $D^k(f^{-1}(x))$? I would be most interested in a non-recursive formula, if such exists. A combinatorial term seems unavoidable.
Related: The first derivative of the inverse function is very well known, and the second one is also not that difficult to determine.
On
About for HIGHER ORDER DERIVATIVES OF FUNCTIONS PARAMETRICALLY DEFINED, See https://barrel.repo.nii.ac.jp/?action=pages_view_main&active_action=repository_view_main_item_detail&item_id=5025&item_no=1&page_id=13&block_id=135
As requested by the OP, I add a link to a paper containing the formula and the bibliographic data (in Japanese).