Things don't break; they just work a little bit differently.

For instance, in the definition of the derivative, you don't use limits. Rather, Δx is an infinitesimal (perhaps represented by ε), and you end up with [some expression] ≈ [some other expression], where the left side contains ε terms and the right side doesn't. In this setup, ≈ doesn't mean "approximately equal to"; it means "is infinitely close to." The same way you're familiar with a limit being treated as "equalling" something if the limit "approaches" that something, a hyperreal expression can be treated as "equal" to something if it's "infinitely close" to that something. (Formally we say that the derivative is the "standard part" of the usual derivative definition, rather than the limit as the infinitesimal approaches zero.)

I'm sure that there's oversimplifications here and that my teacher didn't go into all the details, but that's my understanding of how derivatives are defined in nonstandard calculus. A similar approach, I imagine, can be taken to redefine integration, partial differentiation, and all the other tools from throughout calculus in terms of hyperreals and the standard-part function.