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
![{\displaystyle \Delta y=f'(x)\,\Delta x+\varepsilon \,\Delta x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8cf745b293cc9135d893f5974df49f02e70a6bb2)
for some infinitesimal ε, where
![{\displaystyle \Delta y=f(x+\Delta x)-f(x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67c5d98ee2297f3feca2ec5bd08c5713f7a0ef15)
If
then we may write
![{\displaystyle {\frac {\Delta y}{\Delta x))=f'(x)+\varepsilon ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5783d7ef3cd70e6a2bff094cca9f32fb9e03976)
which implies that
, or in other words that
is infinitely close to
, or
is the standard part of
.
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8cf745b293cc9135d893f5974df49f02e70a6bb2)
holds with the same definition of Δy, but instead of ε being infinitesimal, we have
![{\displaystyle \lim _{\Delta x\to 0}\varepsilon =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6acbd5823f0d8f3f9718bc5573800e8c511a526)
(treating x and f as given so that ε is a function of Δx alone).