How do we take repeated derivatives of composed multivariate functions? for one-dimensional functions, the common tools consist of the Fa\'a di Bruno formula with Bell polynomials; while there are extensions of the Fa\'a di Bruno formula, there are no corresponding Bell polynomials. In this paper, we generalize the single-variable Bell polynomials to take vector-valued arguments indexed by multi-indices which we use to rewrite the Fa\'a di Bruno formula to find derivatives of $\textbf{ f}(\textbf{ g}(\textbf{ x}))$.
Categories: math.CA