I don't know about you, but when I took freshman calculus, it was a big disappointment. Do you find a nice, crystalline, ultra-elegant framework, like the 5 axioms of Euclid?

No, you learn about this random grab-bag of methods for finding the derivatives of all kinds of functions. Every time you want to prove a differentiation formula, it requires a separate, new, creative act of genius. You are wondering why math has suddenly become as ad hoc as chemistry.

Even that wouldn't be so bad, if it were actually useful for anything. But when you start taking advanced physics classes, you find that physicists don't do calculus at all like your freshman calculus class. You've been told that dy/dx isn't really division, dy and dx aren't really numbers, and you basically shouldn't even think about using them that way.....and then you find yourself looking at a chalkboard full of equations where the professor is doing just that.

Hasn't anybody figured out an elegant way of putting this all together?

Well, they actually have, and its dual numbers. I don't know why we aren't just taught this in freshman calculus. Using dual numbers, you can write down proofs of every derivative formula taught in freshman calculus on the front and back of a single piece of paper. You could even do it on one side of the paper, if you used a small, but not microscopic, font.

What's more, after you sit down and write out that piece of paper, you don't even need it anymore :-) You no longer have to memorize dozens of ad hoc differentiation formulas, because if you ever need a formula, you can just derive it yourself--faster than you can pull down a book from the shelf and look the answer up. What's more your brain is now all primed with a good grasp of the structure of the problem you are trying to solve.

And that's only the first few things in the dual-number toybox. You can use them to represent generalized angles (an angle + an orthogonal displacement) and to analyze mechanical linkages (wanna make a walking robot?). You can combine them with vector analysis to make a wonderfully elegant projective geometry which unifies homogeneous coordinates, barycentric coordinates to create an all-but-perfect language for 3d geometry.

What's more it just takes a few days of playing around with them and you are a pro. No big year-long time commitment. Highly recommended.