Unable to look inside to see the textbook content, I went to the Bartlett Publishing website where it’s listed along with alternative textbooks debunking evolutionary theory and fossil forensics. Sorry, I’ll pass.

You can read the two arxiv papers if you like. The book doesn’t really have any technical content that isn’t in those; just more elaboration and exercises pitched at students coming to the subject for the first time.

I certainly wouldn’t recommend this as a sole introductory textbook; it’s obviously produced on a shoestring budget (a LaTeX file with the default template sent over to a print shop) and is somewhat limited in many aspects compared to established calculus textbooks. But the pedagogical idea of focusing on differentials seems sound. I haven’t ever taught an introductory calculus course, but I think it seems entirely plausible that this approach would save some confusion for many students (this is something which could be tested empirically, if any math-ed researcher has the time and budget for a study).

It’s not clear why the guy’s ideas about evolutionary theory (which I know nothing about) have anything to do with his ideas about derivatives. From what I understand he’s a computer programmer and math teacher without extensive training in biology; I wouldn’t expect him to have any insight into evolutionary theory.