Increment Theorem

In nonstandard analysis, the increment theorem states that if a function y equals f(x) is differentiable at x and an infinitesimal change in x is given, then the resulting infinitesimal change in y equals the derivative of f at x times that change plus an infinitesimal error term, which implies the ratio of the two changes is infinitely close to the derivative itself. A parallel version holds in standard calculus for a nonzero real, rather than infinitesimal, change in x, where the same equation holds with an error term that tends to zero as the change in x tends to zero. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/