First and Second Derivatives of the $q$-Rationals

TL;DR

This paper derives explicit formulas for the first and second derivatives of $q$-rationals at $q=1$, providing new mathematical identities that deepen understanding of $q$-rational functions.

Contribution

It introduces closed-form identities for derivatives of $q$-rationals at $q=1$, expanding the theoretical framework of $q$-analogs.

Findings

Closed-form identities for first derivatives

Closed-form identities for second derivatives

Enhanced understanding of $q$-rational functions

Abstract

I prove closed-form identities that I discovered for the first and second derivatives of the $q$-rationals at $q = 1$, in which the $q$-rationals are defined as they are in arXiv:1812.00170.