Increment theorem

In nonstandard analysis, a field of mathematics, the increment theorem states the following: Suppose a function y = f(x) is differentiable at x and that Δx is infinitesimal.

for some infinitesimal ε, where

then we may write

which implies that

, or in other words that

is infinitely close to

is the standard part of

A similar theorem exists in standard Calculus.

Again assume that y = f(x) is differentiable, but now let Δx be a nonzero standard real number.

Then the same equation

holds with the same definition of Δy, but instead of ε being infinitesimal, we have

(treating x and f as given so that ε is a function of Δx alone).