Super Derivatives and Super Integrals

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.