The infinitesimal rule of three

The infinitesimal formulation of strong derivative is simple, in fact, every bit as elementary as the pointwise derivative in the traditional non-infinitesimal approach. However, there are differences in the three basic foundations for Infinitesimal Analysis, even in formulating proofs of intuitive results of calculus. Numbers satisfying (Infsml) are called “infinitesimal” and zero is the only real infinitesimal by the archimedean property of the reals. H. J. Keisler’s Axioms provide an elegant solution to Leibniz’ 300 year old question of how one might rigorously develop calculus using infinitesimals. After A. Robinson’s invention of rigorous Infinitesimal Analysis, it soon became clear that his methods could be extended to a spectrum of problems in analysis, as shown already in 1966. The logical equivalence means that a standard function is derivable in the straightforward infinitesimal sense if and only if its difference quotients converge ‘locally uniformly’.