# 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

${\displaystyle \Delta y=f'(x)\,\Delta x+\varepsilon \,\Delta x}$

for some infinitesimal ε, where

${\displaystyle \Delta y=f(x+\Delta x)-f(x).}$

If ${\displaystyle \scriptstyle \Delta x\not =0}$ then we may write

${\displaystyle {\frac {\Delta y}{\Delta x}}=f'(x)+\varepsilon ,}$

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

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

${\displaystyle \Delta y=f'(x)\,\Delta x+\varepsilon \,\Delta x}$

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

${\displaystyle \lim _{\Delta x\to 0}\varepsilon =0}$

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