Introduction

To many students, differential calculus seems like a set of rules to be applied for solving problems such as optimization problems, tangent problems, etc. This really should not be surprising as differential calculus literally is a set of rules for calculating differences. These rules first appeared in Leibniz's 1684 paper Nova methodus pro maximus et minimus, itemque tangentibus, quae nec fractus nec irrationals, quantitates moratur, et singulare pro illi calculi genus (A New Method for Maxima and Minima as Well as Tangents, Which is Impeded Neither by Fractional Nor by Irrational Quantities, and a Remarkable Type of Calculus for This). A translation of this appears in [5, p. 272–80]. As the title suggests, our students' perceptions are not far off. Indeed, Leibniz's differential calculus is very recognizable to modern students and illustrates the fact that this is really a collection of rules and techniques to compute and utilize (infinitesimal) differences. The fact that Leibniz's notation is so modern in appearance, or rather our notation is that of Leibniz, allows these rules to be presented in a typical calculus class. The author has typically done this while covering the differentials section of the course, as the rules are rules for differentials, not derivatives. Doing this reinforces the rules for computing derivatives and introduces the student to the manipulation of differentials that will be necessary in integration.

A bolder approach, which the author has employed, is to replace the typical “limit of difference quotient” derivations with these heuristic arguments and adopt the point of view that is a ratio of infinitesimals.