TITLE: Learning differential calculus through infinitesimals
QUESTION [5 upvotes]: In class, we've studied differential calculus and integral calculus through limits. We reconstructed the concepts from scratch beginning by the definition of limits, licit operations, derivatives and then integrals. But the teacher really did everything to avoid talking about infinitessimals. For instance when we talked about variable changes we had to swallow that for a certain $u=\phi(x)$, $du = \phi'(x)dx$. What bothers me is how much we use them in maths and physics without understanding them. My question is how can I learn to manipulate infinitessimals? I've read an article on the field of infinitessimals but it didn't quite satisfy my curiosity. What literature would you recommend on the subject knowing that I have some background on linear algebra (vector spaces, matrices), algebraic structures (groups, rings, fields) and monovariable calculus.

REPLY [2 votes]: The best calculus textbook based on infinitesimals is Keisler's textbook Elementary Calculus. This opinion is based on our experience teaching calculus with infinitesimals based on the book over the past three years.  We have taught over 250 students by now using this method, yielding better results than parallel groups that did not use infinitesimals, based on exam scores and teacher evaluations at the end of the course.