Teaching the Derivative Without Initial Reliance on Limits: A Cognitive-Constructivist and Algebraic-Geometric Approach

Abstract

This article presents a cognitive-constructivist approach to teaching the derivative, in which the concept of the
derivative is introduced prior to the formal definition of limit. The traditional method is mathematically rigorous; however, at
the initial stage it confronts the learner with several abstract notions at once, including increment, difference quotient, passage
to the limit, and the relationship between a secant and a tangent. In response to this difficulty, the article discusses an algebraic
geometric model that grounds the derivative in familiar structures: the quadratic function, the double root, and the geometric
idea of tangency. At the point (x=a), the tangent is defined by the condition f(x)−Tₐ(x)=(x−a)²Q(x); On the basis of this criterion,
the derivative emerges as the function of tangent slopes, while the classical limit formula is introduced at a later stage. The
proposed approach is based on the author’s long-term pedagogical experience and observations conducted between 1986 and
2025.