On a Generalization of the Newton Derivative via Mapping Functions

This paper introduces a formal generalization of the classical New tonian derivative by incorporating an auxiliary mapping function, ϵ(x), to redefine the limit-based difference quotient. While traditional calculus evaluates rates of change over a linear identity domain, the proposed General Derivative characterizes the sensitivity of a function f relative to a non-linear transformation of its independent variable. We establish the theoretical framework for this operator and demon strate that the standard derivative emerges as a specific realization where ϵ is the identity mapping. Furthermore, through various com plex transcendental cases, we show that this formulation effectively decouples functional complexity from domain geometry. The study concludes by discussing potential extensions into functional analysis, specifically regarding Fr´echet and Gˆateaux differentiability.