The Specular Derivative

TL;DR

This paper introduces the specular derivative, a new generalized derivative, and develops fundamental calculus theorems and properties for it, with applications to differential equations.

Contribution

It presents the concept of the specular derivative and establishes key calculus theorems and properties for this new derivative.

Findings

Established Quasi-Rolle's Theorem for specular derivatives

Proved the Quasi-Mean Value Theorem for specular derivatives

Applied specular derivatives to solve differential equations

Abstract

In this paper, we introduce a new generalized derivative, which we term the specular derivative. We establish the Quasi-Rolles' Theorem, the Quasi-Mean Value Theorem, and the Fundamental Theorem of Calculus in light of the specular derivative. We also investigate various analytic and geometric properties of specular derivatives and apply these properties to several differential equations.