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.
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.