Differentiation turns out to be a deeper subject than most people expect even if you just stick to the ordinary real numbers rather than venturing into things like hyperreals.

I once saw in an elementary calculus book a note after the proof of a theorem about differentiation that the converse of the theorem was also true but needed more advanced techniques than were covered in the book.

I checked the advanced calculus and real analysis books I had and they didn't have the proof.

I then did some searching and found mention of a book titled "Differentiation" (or something similar) and found a site that had scans for the first chapter of that book. It proved the theorem on something like page 6 and I couldn't understand it at all. Starting from the beginning I think I got through maybe a page or two before it got to my deep with my mere bachelor's degree in mathematics level of preparation.

I kind of wish I'd bought a copy of that book. I've never since been able to find it. I've found other books with the same or similar title but they weren't it.