A Limit-Free Algebraic-Geometric Construction of the Derivative with a Foundational Model in the Class of Polynomial Functions

TL;DR

This paper introduces an algebraic-geometric method to define derivatives without initial limits, extending polynomial concepts to elementary functions and deriving classical limits from an algebraic foundation.

Contribution

It develops a limit-free algebraic-geometric framework for derivatives, connecting polynomial tangency to classical calculus through algebraic and linear decomposition methods.

Findings

Algebraic characterization of tangency via double roots.

Existence and uniqueness of derivatives established algebraically for polynomials.

Classical limit representation derived from the algebraic model.

Abstract

This paper presents an algebraic-geometric construction of the derivative developed initially within the class of polynomial functions without introducing limits at the initial stage. Tangency is characterized by an algebraic condition: the difference between a function and a linear approximation has a double root at a given point. On this basis, the derivative is defined as a functional correspondence assigning to each point the slope of the tangent. Within the class of polynomials, the existence, uniqueness, and fundamental rules of differentiation are established purely algebraically. The constructed model is then extended conceptually to elementary functions and connected to the linear decomposition of functions, from which the classical limit representation of the derivative naturally emerges. Thus, the limit appears not as a starting point but as an analytic expression of an already constructed concept.