Recursive multivariate derivatives of $e^{f(x_1,\dots, x_n)}$ of arbitrary order

Abstract

High-order derivatives of nested functions of a single variable can be computed with the celebrated Fa\`a di Bruno's formula. Although generalizations of such formula to multiple variables exist, their combinatorial nature generates an explosion of factors, and when the order of the derivatives is high, it becomes very challenging to compute them. A solution is to reuse what has already been computed, which is a built-in feature of recursive implementations. Thanks to this, recursive formulas can play an important role in Machine Learning applications, in particular for Automatic Differentiation. In this manuscript we provide a recursive formula to compute multivariate derivatives of arbitrary order of $e^{f(x_1,\dots,x_n)}$ with respect to the variables $x_i$. We note that this method could also be beneficial in cases where the high-order derivatives of a function $f(x_1,\dots,x_n)$ are hard to compute, but where the derivatives of $\log(f(x_1,\dots,x_n))$ are simpler.