Calculus Without Limits : : An Algebraic-Geometric Construction of the Derivative

Davit Kapanadze

ISBN 13: 9798823096607

Published by Authorhouse UK, 2026

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nach der Bestellung gedruckt Neuware - Printed after ordering - This paper presents an algebraic and geometric-functional approachto introducing the derivative for elementary functions without usinglimits. The derivative is defined as a functional correspondencebetween the abscissa of a point on the graph of a function andthe slope of the unique tangent line drawn at that point (the X-Kcorrespondence). The method is developed systematically startingfrom single-variable polynomial functions by introducing the notionsof multiple roots and tangency through an algebraic condition ofrepeated intersection. On this foundation, the derivative functionis constructed and key differentiation rules are established,including the sum, product, quotient, and composite functionrules. The approach is then extended to rational power functions,exponential functions, logarithmic functions with an arbitrary base,and trigonometric functions, yielding the same derivative formulasas in classical analysis. Finally, the increment of a function and thedifferential are interpreted geometrically via the tangent line, andthe classical limit definition of the derivative arises as an analyticalformalization of this geometric differential. The results demonstrateboth mathematical consistency and strong pedagogical potentialfor secondary and undergraduate instruction.

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Bibliographic Details

• Title: Calculus Without Limits : : An Algebraic-Geometric Construction of the Derivative
• Publisher: Authorhouse UK
• Publication Date: 2026
• Language: English
• ISBN 13: 9798823096607
• Binding: Taschenbuch
• Condition: Neu
• Dimensions: 229W x 152H x 4D millimeters 229 millimeters width by 152 millimeters height by 4 millimeters depth
• Weight: 95 grams

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Synopsis

This paper presents an algebraic and geometric–functional approach to introducing the derivative for elementary functions without using limits. The derivative is defined as a functional correspondence between the abscissa of a point on the graph of a function and the slope of the unique tangent line drawn at that point (the X–K correspondence). The method is developed systematically starting from single-variable polynomial functions by introducing the notions of multiple roots and tangency through an algebraic condition of repeated intersection. On this foundation, the derivative function is constructed and key differentiation rules are established, including the sum, product, quotient, and composite function rules. The approach is then extended to rational power functions, exponential functions, logarithmic functions with an arbitrary base, and trigonometric functions, yielding the same derivative formulas as in classical analysis. Finally, the increment of a function and the differential are interpreted geometrically via the tangent line, and the classical limit definition of the derivative arises as an analytical formalization of this geometric differential. The results demonstrate both mathematical consistency and strong pedagogical potential for secondary and undergraduate instruction.