Divided differences of implicit functions

Divided differences can be viewed as a discrete analogue of derivatives and are commonly used in approximation theory, see [1] for a survey. Recently, the second author and Lyche established two univariate chain rules for divided differences [2], both of which can be viewed as analogous to Faà di Bruno’s formula for differentiating composite functions [6,7]. One of these formulas was simultaneously discovered by Wang and Xu [5]. In a follow-up preprint, the other chain rule was generalized to the composition of vector-valued functions of several variables [4], yielding a formula analogous to a multivariate version of Faà di Bruno’s formula [8]. In [3], the univariate chain rule was applied to find a formula for divided differences of the inverse of a function. In Theorem 1, the Main Theorem of this paper, we use the multivariate chain rule to prove a similar formula for divided differences of implicitly defined functions. Equation 16 shows that the formula for divided differences of inverse functions in [3] follows as a special case. More precisely, let y be a function that is defined implicitly by a function g : R2 → R via g (