  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.
