A Limit-Free Algebraic-Geometric Construction of Derivatives for Elementary Functions

TL;DR

This paper introduces a limit-free, algebraic-geometric method for constructing derivatives of elementary functions, connecting geometric intuition with classical calculus without initial reliance on limits.

Contribution

It extends previous work on polynomial derivatives to elementary functions using a geometric approach, avoiding the traditional limit process.

Findings

Derivatives are constructed via geometric interpretation without limits.

Classical differentiation formulas emerge naturally from the framework.

The approach provides a conceptual bridge between geometry, algebra, and calculus.

Abstract

This paper continues the author's previous work on a limit-free algebraic-geometric construction of the derivative in the class of polynomial functions and extends the proposed framework to elementary functions. Derivatives of rational power, exponential, logarithmic, trigonometric, and inverse trigonometric functions are constructed through the geometric interpretation of the tangent line, inverse symmetry, and local linear structure, without treating the limit as the initial defining mechanism. Within the proposed approach, the derivative is introduced from the outset as a functional correspondence assigning to each point the slope coefficient of the tangent line. The paper demonstrates that the classical differentiation formulas arise naturally from interconnected geometric and algebraic structures and are subsequently consistent with standard limit-based analysis. From a methodological perspective, the study proposes the logical sequence: Tangent, Local Linear Structure, Limit formalisation. Thus, the paper presents a conceptual bridge between geometric intuition, algebraic construction, and classical differential calculus.