this is from my book:

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

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

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

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