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