Increment theorem

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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 \scriptstyle\Delta x\not=0 then we may write

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

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

See also[edit]