Explicit expressions for a family of Bell polynomials and derivatives of some functions

TL;DR

This paper derives explicit formulas for derivatives of elementary functions, including the arc sine, and uses these to obtain expressions for a family of Bell polynomials related to the square function.

Contribution

It introduces new explicit formulas for derivatives and Bell polynomials, expanding tools for analyzing derivatives of elementary functions.

Findings

Explicit formulas for derivatives of arc sine

Explicit expressions for Bell polynomials related to the square function

Applications to derivatives of elementary functions

Abstract

In the paper, the authors first inductively establish explicit formulas for derivatives of the arc sine function, then derive from these explicit formulas explicit expressions for a family of Bell polynomials related to the square function, and finally apply these explicit expressions to find explicit formulas for derivatives of some elementary functions.