A direct and elementary derivation of the multivariate Faà di Bruno formula

By applying Taylor expansion and the Leibniz rule for partial derivatives, we provide a straightforward deduction of the multivariate Faà di Bruno formula, yielding a concise and simple expression for the derivatives of the composite function $g\circ \f :\mathbb{R}^n\to\mathbb{R}$, with $\f :\mathbb{R}^n\to\mathbb{R}^m$ and $g :\mathbb{R}^m\to\mathbb{R}$. Our formulation extends the original Faà di Bruno formula to the multidimensional case in a natural way.