I only recently became aware that it is possible to formalize the methods that Newton and Leibniz first used when developing calculus. In fact, not only is it possible, but it is quite easy to understand, and the notation is more elegant than those of limits.
I had always looked down on the pseudo-proofs I was shown in physics lectures, handwavy sloppy arguments that defied the rigour and precision that centuries of mathematics fought for. I tolerated them because I needed the results. But it seems with a few definitions, you can have your cake and eat it.
One introduction to this subject is a free online book: “Elementary Calculus: An Approach Using Infinitesimals” by H. Jerome Keisler.