Derivatives of composite functions when component functions are power, log or exponential

Abstract

Faa di Bruno gave a chain rule for the derivatives of a function of a function. It is succinctly expressed in terms of Bell polynomials. This hasa particularly simple form when the derivatives of the "inner" function are proportional to a geometric series. We call such functions geometric. Examples include power, exponential, log, and 1/(1-x). Extensions aregiven to compositions of more than two functions, such as finding the derivatives of the log-normal density. Simple formulas are given for the derivatives of any composition of linear, power, log and exponential functions. Wealso give new results for Bell polynomials and Stirling numbers, and extensions of the Hermite polynomials.