On Higher Partial Derivatives of Implicit Functions and their Combinatorics

TL;DR

This paper derives two explicit formulas for higher-order partial derivatives of implicit functions, analyzing their combinatorial structures and providing tools for advanced calculus involving multiple variables.

Contribution

It introduces two novel closed-form expressions for derivatives of implicit functions and explores their combinatorial properties, enhancing understanding of their structure.

Findings

Two explicit formulas for derivatives of implicit functions.

Analysis of the combinatorial coefficients in the formulas.

Comparison of formulas based on different combinatorial structures.

Abstract

We consider a scalar-valued implicit function of many variables, and provide two closed formulae for all of its partial derivatives. One formula is based on products of partial derivatives of the defining function, the other one involves less products of building blocks of multinomial type, and we study the combinatorics of the coefficients showing up in both formulae.