Approximation formula for Higher-order derivative

Question

I have this assigment

Derive by using Taylor approximation of $f$ a formula for approximation of $f′(x_0),f′′(x_0),f′′′(x_0),f′′′′(x_0)$ with an error term of order $h^4$.

I have already done $f′(x_0)$ and $f′′(x_0)$ but I'm stuck trying to get $f'''(x)$. I've done the Taylor expansion for $f(x+h),f(x-h),f(x+2h),f(x-2h),f(x+3h),f(x-3h)$ and don't know what else to do.

Can someone give me some advice o help me to get $f'''(x)$?

PD Excuse my bad English.

EDIT:

What the assigment want me to get is something like

$f'''(x)= \frac{-f(x+3h)+8f(x+2h)-13f(x+h)+13f(x-h)-8f(x-2h)+f(x-3h)}{8h^3}+O(h^4)$

Answer 1

Hint: read Section 1.2 of this text.