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 $Q_i$.
$$Q_i=p (P_{}-P_i)/(u_{i+p+1}-u_{i+1})$$
I think the built-in `BSplineFunction[2Dpts]'` just returns the non-rational curve.

For the rational curve:
$$C^w(u)=\frac{\sum_i=0^n N_{i,p}w_iP_i}{\sum_i=0^n N_{i,p}w_i}=\frac{A(u)}{w(u)}$$

$${C^w}'(u)=\frac{A(u)}{w(u)}=\frac{A'(u)w(u)-A(u)w'(u)}{w^2(u)}$$

I will write a full answer when I have a laptop.Now I just using a smart-phone.⊙▽⊙