Super Derivatives and Super Integrals
Description
We introduce a unified framework for limit-difference quotients based on auxiliary domain mappings. By replacing the classical linear increment with a nonlinear deformation mapping, we define a class of operators that measure functional variation relative to transformed scales. This formulation naturally encompasses the classical deriva tive, directional and Fr´echet derivatives, as well as nonlocal difference operators arising in quantum calculus and (p,q)-analysis. The proposed framework distinguishes between local differential behavior and nonlocal, scale-dependent sensitivity, providing a sys tematic interpretation of secant-type operators within a common limit structure. Several analytical examples are presented to illustrate the behavior of the operator for smooth, nonsmooth, and nonlocal config urations. Extensions to Banach spaces and dual-mapping settings are also discussed, highlighting the relevance of the approach to functional analysis and nonlocal calculus.