# Rob Ely

11.06.2021

**Teaching Calculus with Infinitesimals and Differentials**

**Keywords:** Calculus; differentials; infinitesimals; nonstandard analysis; definite integral

**Abstract:** Several new approaches to calculus in the U.S. have been studied recently that are grounded in infinitesimals or differentials rather than limits. These approaches seek to restore to differential notation the direct referential power it had during the first century after calculus was developed. In these approaches, a differential equation like dy = 2 xdx is a relationship between increments of x and y , making dy / dx an actual quotient rather than code language for the limit of (1/h)[ f ( x + h ) - f ( x )] as h approaches 0. A definite integral of 2 x dx is a sum of pieces of the form 2 x dx , not the limit of a sequence of Riemann sums. In this article I motivate and describe some key elements of differentials-based calculus courses, and I summarize research indicating that students in such courses develop robust quantitative meanings for notations in single- and multi-variable calculus. (Click here for article)