Calculus Without Limits

: An Algebraic-Geometric Construction of the Derivative

von Davit Kapanadze

54 Seiten, Taschenbuch

Kurzbeschreibung des Verlags

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.

• Themen: Mathematik und Naturwissenschaften Mathematik Algebra
• ISBN: 9798823096607
• Sprache: Englisch
• Erscheinungsdatum: 26.04.2026
• Größe: 229 x 152 mm
• Verlag: AuthorHouse UK