Let me first state that I too am no expert (which is why I am asking people in sf).
As far as I know, the concept of "infinitisimal" has been abandoned since Weierstrass gave a rigorous definition of the concept of limits and used limits to "derive" calculus.
The exception is, as DrRocket says, the non-standard calcuclus, which is, as far as I know, also a rigorous method worked out by a guy in the 60's.
I think the problem is that dy/dx is really a notation and not a ratio.
The notation is sometimes abused by (e.g.) multiplying by dx on both sides, which has no meaning from the dy/dx limit definition.
My question is, how can you justify doing it anyways?
P.S.
If you like the infinitisimals, I can recommend "Calculus made easy" by Silvanus Thompson. It's very informal, only relying on algebra.