Chain rules for higher derivatives

Huang-Nan Huang, Stefania Marcantognini and Nicholas Young

Keywords

Faa di Bruno's formula, directional derivatives

Status

To appear in Mathematical Intelligencer

Abstract

We define a notion of higher-order directional derivative of a smooth function and use it to establish three simple formulae for the nth derivative of the composition of two functions. These three "higher-order chain rules" are alternatives to the classical formula of St Francis Faa di Bruno. Our formulae are less explicit than Faa di Bruno's formula, but are simpler and avoid the need for combinatorial or Diophantine considerations.

The paper is available in pdf format here

Nicholas Young
27 September 2005.