On derivations with respect to finite sets of smooth functions

TL;DR

This paper investigates conditions under which derivations with respect to finite sets of smooth functions are necessarily standard derivations, focusing on specific functions like exponential, trigonometric, and hyperbolic functions.

Contribution

It characterizes finite sets of differentiable functions that ensure derivations are automatically standard, extending understanding of derivation structures.

Findings

Derivations of product functions and certain elementary functions are standard.

Finite sets of differentiable functions can determine when derivations are standard.

The paper provides conditions for derivations to be automatically standard based on the functions involved.

Abstract

The purpose of this paper is to show that functions that derivate the two-variable product function and one of the exponential, trigonometric or hyperbolic functions are also standard derivations. The more general problem considered is to describe finite sets of differentiable functions such that derivations with respect to this set are automatically standard derivations.