First derivative ($f_0 = f(x_0)$, $f_{-h}=f(x_0-h)$ and $f_{+h}=f(x_0+h)$ $$\frac{f_h-f_{-h}}{2h}=f'_0+\sum_{j=1}^{\infty}\frac{f_0^{(2j+1)}}{(2j+1)!}h^{2j}.$$ Second derivative $$\frac{ f_h -2f_0 +f_{-h}}{h^2}=f_0''+2\sum_{j=1}^{\infty}\frac{f_0^{(2j+2)}}{(2j+2)!}h^{2j}.$$