Also (in case anyone's still looking at this thread) the accuracy is a separate issue from numerical stability. That simple quadrature rules like trapezoid can

Also (in case anyone's still looking at this thread) the accuracy is a separate issue from numerical stability. That simple quadrature rules like trapezoid can be so accueate (error is geometric, not just algebraic to high order) is because of the smoothness of the integrand and periodicity. An N-point high-order estimate of the derivative along the real line, using finite differences, would not have these properties.