Exercise 5.6.24 from Arfken - Calculate y'_0

[thesaruman]

Homework Statement

Exercise 5.6.24 from Arfken's Essential Mathematical Methods for Physicists. You have a function tabulated at equally spaced values of the argument:
$$\left\{ \begin{array}{c} y_n=y(x_n)\\x_n = x+nh \end{array}\right.$$
Show that the linear combination
$$\frac{1}{12h}\left\{ -y_2+8y_1-8y_{-1}+y_{-2} \right\}$$
yields
$$y'_0-\frac{h^4}{30}y^{(5)}_0+\cdots .$$

Homework Equations

I used:
$$\frac{d^ny_0}{dx^n}=\sum_{m=0}^n(-1)^n\left( \begin{array}{c} n\\ m \end{array} \right)y_{\frac{n}{2}-m}$$ for the derivatives of an even order and a geometric average between the backward and forward derivatives for the odd ones.
Keep in mind that $$y(x)=y_0$$, $$y(x+h)=y_1$$,...
Also used the Taylor expansion of y(x+h):
$$f(x+h)=\sum_{m=0}^\infty \frac{h^n}{n!}f^{(n)}(x)$$.

3. The Attempt at a Solution

I solved the Taylor expansion for $$y'_0$$ and tried to modify the result accordingly with the equation the author expressed. But there was no way to get it fit. I guess I didn't understand how the author calculates the derivative of the function.
Please, someone help. I am getting crazy with the exercises of this book.

 

[thesaruman]

Well, I finally got the answer and it was very simple in fact!
It's only necessary to expand all the terms enclosed in the parenthesis and truncate the expansion in 5^th order.