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.