Increment theorem
# 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. Then

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

for some infinitesimal ε, where

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

If Δx≠0{\textstyle \Delta x\neq 0} then we may write

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

which implies that Δy/Δx $≈f^′(x){\textstyle$ {\frac {\Delta y}{\Delta x}}\approx f'(x)}, or in other words that Δy/Δx {\textstyle {\frac {\Delta y}{\Delta x}}} is infinitely close to $f^′(x){\textstyle$ f'(x)}, or $f^′(x){\textstyle$ f'(x)} is the standard part of Δy/Δx {\textstyle {\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).

