Some Problems in Differentiation

TL;DR

This paper discusses methods for computing higher-order derivatives of functions given parametrically or implicitly, including historical context and mathematical techniques, contributing to the understanding of differentiation in complex function scenarios.

Contribution

It introduces approaches for calculating nth derivatives of parametric and implicit functions, with historical insights and references to related work.

Findings

Provides formulas for nth derivatives of parametric functions

Extends techniques to implicit functions

Includes historical perspective on differentiation methods

Abstract

We compute the nth derivative of a function given parametrically, and of one given implicitly, and some history for both problems. I am posting this version of the paper at the request of Shaul Zemel, whose forthcoming paper The Combinatorics of Higher Derivatives of Implicit Functions refers to it.