Post Made Community Wiki by S. Carnahan
1

I hope my answer is read as a response to the question asked, rather than as either a defense of or disagreement with the choices the pedagogists (is that a word?) make.

I think one of the main reasons to teach derivatives in terms of the $h\to 0$ limit is that it captures the dual notions of "instantaneous velocity" and "slope", which are respectively physical and geometric.

(Ok, now I will mention some personal opinions about teaching calculus. I love physics, and sometimes pretend to be a physicist, so for me the geometric/physical meanings of calculus are very important. So I would love if they were emphasized more. Unfortunately, we do not do enough in introductory calculus classes in that direction, and it is very hard to present functions and ask students to find the slopes of their graphs without essentially teaching them these black-box techniques. So I don't know whether it's worth it: maybe we should just do the algebraic part of calculus --- it's the only thing we tend to test anyway. I also don't really think that MO is the best place to get into that discussion, though, and I don't think that OP intended as such.)