Formal differential variables and an abstract chain rule

TL;DR

This paper introduces a formal framework for iterated differentials and a multivariable chain rule that extends to higher derivatives, addressing limitations of the classical chain rule.

Contribution

It presents a novel approach to iterated differentials and a finite calculus chain rule, expanding the theoretical tools for higher-order calculus.

Findings

Developed a formal theory of iterated differentials

Proposed a multivariable chain rule for higher derivatives

Challenged the notion that finite calculus lacks a chain rule

Abstract

One shortcoming of the chain rule is that it does not iterate: it gives the derivative of f(g(x)), but not (directly) the second or higher-order derivatives. We present iterated differentials and a version of the multivariable chain rule which iterates to any desired level of derivative. We first present this material informally, and later discuss how to make it rigorous (a discussion which touches on formal foundations of calculus). We also suggest a finite calculus chain rule (contrary to Graham, Knuth and Patashnik's claim that "there's no corresponding chain rule of finite calculus").