You could think of it as a short hand for a more convoluted argument
using finite differences, some regularity conditions and limiting processes.
One thing which makes me very angry is that textbooks treat quantities such as "dy" and "dx" as numbers, yet they are not. When they come to solve differencial equation the authors simply say divide both sides by dx or something like that, it lacks mathematical rigor. Futhermore, they give "dy" and "dx" seperate meanings, but the problems it that they only make sense as dy/dx. And the only reason why we use these symbols is because this Leibniz notation sometimes is easier to use. I was reading an article on www.wikipedia.com about this topic and they said that the symbols "dy" and "dx" in fact do have seperate meanings! They explained that it uses new types of concepts known as "infinitesimal numbers" these numbers are not real. With these numbers we can construct a rigorus meaning to the differencial. This type of construction forms "non-standard analayis" because in a sense it is still analysis but without the theory of limits. Do you agree that textbooks do not give a rigorous definition for a differencial? It is perhaps I am not using an advanced textbook, they probably address this issue, please help.