How to stare at the higher-order n-dimensional chain rule without losing your marbles

Abstract

Given two real functions on the real line f and g, the Faa di Bruno provides the higher order derivative of the composition of f and g, as a summation over the lower order derivatives of f and g individually. The corresponding multi-dimensional generalization is substantially more difficult due to the complicated combinatorial considerations one must take into account when dealing with standard multi-indices. In this note we provide a simple statement and derivation of the multi-dimensional Faa di Bruno formula by resorting to notions from multiset theory.