Divided differences of inverse functions and partitions of a convex polygon

We derive a formula for an n-th order divided difference of the inverse of a function. The formula has a simple and surprising structure: it is a sum over partitions of a convex polygon with n+1 vertices. The formula provides a numerically stable method of computing divided differences of k-th roots. It also provides a new way of enumerating all partitions of a convex polygon of a certain type, i.e., with a specified numbers of triangles, quadrilaterals, and so on, which includes Catalan numbers as a special case.