Note that the improved accuracy in the evaluation of the derivatives gives a better error approximation, $O(h^5)$ vs.\ $O(h^3)$ . But this is again the local error approximation. Using Simpson's rule we can easily compute the integral of Eq. (1) to be $$\begin{equation} I=\int_a^bf(x) dx=\frac{h}{3}\left(f(a) + 4f(a+h) +2f(a+2h)+ \dots +4f(b-h)+ f_{b}\right), \tag{6} \end{equation}$$ with a global error which goes like $O(h^4)$.