Chapter 5.1, Problem 19E

Prove the second-order formula for the third derivative

${{f^{\prime }}^{\prime }}^{\prime }\left(x\right)=\frac{-f\left(x-2h\right)+2f\left(x-h\right)-2f\left(x+h\right)+f\left(x+2h\right)}{2h^3}+O\left(h^2\right)$.