Increment theorem

In nonstandard analysis, a field of mathematics, the increment theorem states the following: Suppose a function {{math|1=y = f(x)}} is differentiable at {{mvar|x}} and that {{math|Δx}} is infinitesimal. Then

$$\backslash Delta\; y\; =\; f\text{'}(x)\backslash ,\backslash Delta\; x\; +\; \backslash varepsilon\backslash ,\; \backslash Delta\; x$$

for some infinitesimal {{mvar|ε}}, where

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

If $\backslash Delta\; x\; \backslash neq\; 0$ then we may write

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

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

A similar theorem exists in standard Calculus. Again assume that {{math|1=y = f(x)}} is differentiable, but now let {{math|Δx}} be a nonzero standard real number. Then the same equation

$$\backslash Delta\; y\; =\; f\text{'}(x)\backslash ,\backslash Delta\; x\; +\; \backslash varepsilon\backslash ,\; \backslash Delta\; x$$

holds with the same definition of {{math|Δy}}, but instead of {{mvar|ε}} being infinitesimal, we have

$$\backslash lim\_\{\backslash Delta\; x\; \backslash to\; 0\}\; \backslash varepsilon\; =\; 0$$

(treating {{mvar|x}} and {{math|f}} as given so that {{mvar|ε}} is a function of {{math|Δx}} alone).