Increment theorem

In non-standard 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. Then

for some infinitesimal ε, where

If Δ x ≠ 0 then we may write

which implies that Δ y Δ x ≈ f ′ ( x ) , or in other words that Δ y Δ x is infinitely close to f ′ ( x ) , or f ′ ( x ) is the standard part of Δ y Δ x .