# 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

$\Delta y=f'(x)\,\Delta x+\varepsilon \,\Delta x$ for some infinitesimal ε, where

$\Delta y=f(x+\Delta x)-f(x).$ If $\Delta x\not =0$ then we may write

${\frac {\Delta y}{\Delta x}}=f'(x)+\varepsilon ,$ which implies that ${\frac {\Delta y}{\Delta x}}\approx f'(x)$ , or in other words that ${\frac {\Delta y}{\Delta x}}$ is infinitely close to $f'(x)$ , or $f'(x)$ is the standard part of ${\frac {\Delta y}{\Delta x}}$ .

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

$\Delta y=f'(x)\,\Delta x+\varepsilon \,\Delta x$ holds with the same definition of Δy, but instead of ε being infinitesimal, we have

$\lim _{\Delta x\to 0}\varepsilon =0$ (treating x and f as given so that ε is a function of Δx alone).