I think you've missed the point. You're correct that one can do basic calculus using a bunch of ad-hoc shortcuts, and it is commonly taught this way. Also, as y

I think you've missed the point. You're correct that one can do basic calculus using a bunch of ad-hoc shortcuts, and it is commonly taught this way. Also, as you probably know, calculus was first formalised using limits not infinitesimals. However this theory of infinitesimals is not ad-hoc. It is a complete and rigorous alternate formulation of calculus in terms of an extension of the reals, known as the hyperreals, that includes infinitesimals. This allows computation in a principled way that matches the intuitions on which beginning calculus is often taught. It's pretty cool stuff, and worth reading about even if you never use it in practice.