The Derivatives of the Inverse of a One-to-One Function

The derivatives of the inverse of a one-to-one function are needed in a range of applied contexts, from random variate generation and molecular simulation to nuclear physics and bias reduction for maximum likelihood estimates, yet existing treatments derive them by ad hoc, application-specific means without a unifying framework. Here, we give the general derivative of the inverse of a one-to-one function, firstly by a recurrence formula, secondly by repeated differentiation, and thirdly—and most explicitly—in closed form using the partial exponential Bell polynomials associated with Faà di Bruno’s chain rule, providing a single representation that subsumes and extends earlier case-specific results and that can be taken to arbitrary order. We illustrate the practical value of these results in mathematical statistics, applying them to bias reduction for maximum likelihood estimates in one-parameter exponential families, including the gamma shape parameter and canonical regression models. Python programs implementing the recurrence and the Bell polynomial representations are included.