Differentials can be done rigorously simply using variables, as Cauchy showed us 200 years ago. This is done in Stewart (standard undergrad textbook), in additi

Differentials can be done rigorously simply using variables, as Cauchy showed us 200 years ago. This is done in Stewart (standard undergrad textbook), in addition to the limit/derivative approach and he freely uses them in proofs and derivations throughout the book when they are more convenient. As the article discusses, the difficulty in treating differentials algebraically is how higher order derivatives behave. This problem exist in Robinson's nonstandard analysis as well, and the authors of the paper point this out at the end of section 2, along with examples in the literature. The substance of the article is improving the notation of differentials so that the actual algebraic properties of e.g. the chain rule appear sensible.