Teaching Differentiation: A Rare Case for the Problem of the Slope of the Tangent Line

TL;DR

This paper addresses a common student misconception about derivatives, demonstrating that an undefined derivative expression does not necessarily mean the function is non-differentiable, and offers classroom-effective methods for such cases.

Contribution

It introduces a methodological approach to handle functions with undefined derivative expressions at certain points, clarifying misconceptions in calculus education.

Findings

Functions can be differentiable even if their derivative expressions are undefined.

Standard derivative calculation methods may miss solutions due to undefined expressions.

A classroom-friendly method for analyzing such functions is proposed.

Abstract

In this article we discuss an important students' misconception about derivatives, that the expression of the derivative of the function contains the information as to whether the function is differentiable or not where the expression is undefined. As a working example we consider a typical Calculus problem of finding the horizontal tangent lines of a function. Following the standard procedure, we derive the expression for the derivative using Product Rule. The search for the values of the independent variable, that make the derivative equal zero, leads to missing the unique solution of the problem. We show that in this case, even though the expression of the derivative is undefined, the function indeed possesses the derivative at the point. We also provide the methodological treatment of such functions, which can be effectively used in the classroom.