Let $\vec{U}=\{u_0,u_1,\ldots,u_m\}$ denotes a non-decreasing sequence of real numbers, i.e, $u_i\leq u_{i+1} \quad i=0,1,2\ldots m-1$.

and the $i$-th B-spline basis function of $p$-degree, denoted by $N_{i,p}(u)$, is defined as below:

$$ \begin{cases} N_{i,0}(u)= \begin{cases} 1 & u_i\leq u<u_{i+1}\\ 0 & otherwise \end{cases}\\ N_{i,p}(u)= \frac{u-u_i}{u_{i+p}-u_i}N_{i,p-1}(u)+\frac{u_{i+p+1}-u}{u_{i+p+1}-u_{i+1}}N_{i+1,p-1}(u) \end{cases} $$

By reading the textbook The NURBS Book, I can know the following recursive formula about the derivetive of $N_{i,p}(u)$. Namely,

$$ \frac{d}{du}N_{i,p}(u)=p\left[ \frac{N_{i,p-1}(u)}{u_{i+p}-u_i}-\frac{N_{i+1,p-1}(u)}{u_{i+p+1}-u_{i+1}} \right] \qquad (1) $$

In addition, I can also understand the verification process by mathematical induction that the author given in the textbook. However, I would like to know where this formula came from. Namely,

• How to deduce the derivative formmula of the B-spline basis function $N_{i,p}(u)$?

• Although the author has given the reference The Computation of all the Derivatives of a B-spline Basis in the Bibiology, I cannot download that paper by the libriary of our university. In addition, the reference just for another recursive formula(see Eq.(2)), not for Eq.(1)

$$ N_{i,p}^k=\frac{p}{p-k}\left(\frac{u-u_i}{u_{i+p}-u_i}N_{i,p-1}^k+\frac{u_{i+p+1}-u}{u_{i+p+1}-u_{i+1}}N_{i,p-1}^k\right) \quad (2) $$

where $k=0,1,\cdots,p-1$