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 $${\displaystyle \Delta y=f'(x)\,\Delta x+\varepsilon \,\Delta x}$$ for some infinitesimal ε, where $${\displaystyle \Delta y=f(x+\Delta x)-f(x).}$$

If ${\textstyle \Delta x\neq 0}$ then we may write $${\displaystyle {\frac {\Delta y}{\Delta x}}=f'(x)+\varepsilon ,}$$ which implies that ${\textstyle {\frac {\Delta y}{\Delta x}}\approx f'(x)}$, or in other words that ${\textstyle {\frac {\Delta y}{\Delta x}}}$ is infinitely close to ${\textstyle f'(x)}$, or ${\textstyle f'(x)}$ is the standard part of ${\textstyle {\frac {\Delta y}{\Delta x}}}$.

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