The Combinatorics of Higher Derivatives of Implicit Functions

TL;DR

This paper derives a closed-form combinatorial formula for higher-order derivatives of implicit functions, revealing the underlying combinatorial structure of the coefficients involved.

Contribution

It introduces a novel explicit formula for derivatives of implicit functions, connecting calculus with combinatorial mathematics.

Findings

Provides a closed-form expression for derivatives of implicit functions.

Explains the combinatorial structure of the coefficients in the formula.

Bridges calculus and combinatorics through explicit derivative formulas.

Abstract

We prove a closed formula for the derivative, of any order, of a implicit function, in terms of some binomial building blocks, and explain the combinatorics behind the coefficients appearing in the formula.