Soft derivative

This study presents a comprehensive investigation of soft and upper (lower) soft derivatives within the framework of Molodtsov’s soft set theory, extending it through the introduction of novel and complementary notions: left and right soft derivatives. It rigorously develops the fundamental properties of these concepts, including algebraic and order-theoretic rules, and establishes their relationships with boundedness and soft continuity types. The paper offers insightful geometric interpretations that significantly enhance conceptual understanding. Furthermore, it introduces absolute ε-extrema, investigates their fundamental properties, and characterizes the local (τ, ε)-extrema defined by Molodtsov. This study presents analogs of Rolle’s Theorem for upper and lower soft derivatives and interprets them geometrically with supporting visual illustrations. Moreover, it establishes and geometrically interprets analogs of the Mean Value Theorem for upper and lower soft derivatives. By systematically investigating fundamental concepts in soft analysis and presenting detailed results, the present paper constructs a comprehensive foundation that strengthens the mathematical structure of soft analysis and paves the way for advanced developments, such as soft integrals, soft directional derivatives, and soft gradients.