  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 $\textbf{f}(\textbf{g}(\textbf{x}))$.
