For the non-rational B-spline curve of degree $p$, its derivative is a $p-1$ degree non-rational curve.
where, the new control points is $Q_i$.
$$Q_i=p \frac{P_{i+1}-P_i}{u_{i+p+1}-u_{i+1}}$$
I think the built-in f=BSplineFunction[2D-points]; f' just returns a non-rational curve.
For the rational curve: $$C^w(u)=\frac{\sum_{i=0}^n N_{i,p}(u)w_iP_i}{\sum_{i=0}^n N_{i,p}(u)w_i}=\frac{A(u)}{w(u)}$$
$${C^w}'(u)=\frac{A(u)}{w(u)}=\frac{A'(u)w(u)-A(u)w'(u)}{w^2(u)}$$
I will write a full answer when I have a laptop. Now I just using a smart-phone.