this is from my book:

Suppose that $\displaystyle f\in C^2[a,b]$. Let $\displaystyle \overline{x}\in [a,b]$ be an approximation to $\displaystyle p$ such that $\displaystyle f'(\overline{x})\neq 0$ and $\displaystyle |p-\overline{x}|$ is "small". Consider the first Taylor polynomial for $\displaystyle f(x)$ expanded about $\displaystyle \overline{x}$,

$\displaystyle f(x) = f(\overline{x})+(x-\overline{x})f'(\overline{x})+\frac{(x-\overline{x})^2}{2}f''(\xi(x))$,

where $\displaystyle \xi(x)$ lies between $\displaystyle x$ and $\displaystyle \overline{x}$.

Why do i need to use $\displaystyle \xi(x)$? As far as I know, Newton-Raphsons assumes that $\displaystyle |p-\overline{x}|$ is so small that we can ignore $\displaystyle (x-\overline{x})^2$ and thus also the second derivative.. Someone care to explain?