Summary: A formula for higher order derivatives of composite functions of several variables is given. This formula is a direct generalization of the old formula for the one-variable case given in 1855 by Faà di Bruno. It is expected that the new formula has applications to nonlinear PDE and various other areas in analysis. In §3 of the article a modification of the formula is given. The modified formula is expected to be useful when one solves the Cauchy problem for a nonlinear PDE.