On the Divided Difference Form of Faà di Bruno's Formula

Abstract

The $n$-divided difference of the composite function $h:=f\circ g$ of functions $f$, $g$ at a group of nodes $t_0, t_1, \cdots, t_n$ is shown by the combinations of divided differences of $f$ at the group of nodes $g(t_0), g(t_1), \cdots, g(t_m)$ and divided differences of $g$ at several partial group of nodes $t_0, t_1,\cdots, t_n$, where $m=1, 2,\cdots, n$. Especially, when the given group of nodes are equal to each other completely, it will lead to Faà di Bruno's formula of higher derivatives of function $h$.