[ramblenode]

Solving an ODE by separation of variables. With the limit definition this is just a notational trick, requires additional proof to justify, and confuses students. With infinitesimals, the separation dy = v * dx is a rigorous statement, making the logic of the method immediately obvious.

[credit_guy]

I think Gian-Carlo Rota would disagree with you. See Section 7 of

https://web.williams.edu/Mathematics/lg5/Rota.pdf

[ramblenode]

Rota is complaining that differentials are introduced as an ad hoc technique, seemingly breaking the rules. If differentials are taught from the beginning, i.e. y + dy = f(x + dx) where f'(x) is a convenience function, this is not an issue. Of course, there are other issues with teaching differentials, namely why products of differentials vanish. On the other hand, limits aren't rigorously justified in an introductory course either.