For the non-rational B-spline curve of degree $p$, its derivative is a $p-1$ degree non-rational curve .
I think the buil-tin BSplineFunction[2Dpts]' just returns the non-ratiinal curve.
For the rational curve: $$C^w(u)=\frac{\sum_i=0^n N_{i,p}w_iP_i}{sum_i=0^n N_{i,p}w_i}=\frac{A(u)}{w(u)}$$
$$C^w'(u)=(\frac{A(u)}{w(u)})=\frac{A'(u)w(u)-A(u)w'(u)}{w^2(u)}$$