**Description**

Let $\vec{U}=\{u_0,u_1,\ldots,u_m\}$ denote 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:

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

From reading the textbook The *NURBS* Book, I know the following recursive formula about the derivative of $N_{i,p}(u)$:

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

In addition, I can also understand the *verification process* by mathematical induction that the author gives in the textbook pp.59-60. Namely,

**(1)** Verifying the correctness of this recursive formula for $p=1$;

**(2)** Assuming this formula is right for $p=k$, then proved that this formula is also right for $p=k+1$ with help of the assumption. 

However, I would like to know **where this formula came from**. The author just gives the conclusion and proved it by **mathematical induction**.

**QUESTION**

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

- Although the author has given a related reference [The Computation of all the Derivatives of a B-spline Basis](http://imamat.oxfordjournals.org/content/17/1/15.short) in the bibliography, I cannot download that paper through the library of our university. In addition, the reference is just for another recursive formula (please 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 $k=0,1,\cdots,p-1$

Lastly, I discovered that **Eq.(1)** was more useful than **Eq.(2)**, and it was implemented in Wolfram *Mathematica*. For instance,

    knots = {0, 0, 0, 0, 1/3, 2/3, 1, 1, 1, 1};
    D[BSplineBasis[{3, knots}, 2, x], x]
    (*9/2 BSplineBasis[{2, {0, 0, 0, 0, 1/3, 2/3, 1, 1, 1, 1}}, 2, x] - 
        3 BSplineBasis[{2, {0, 0, 0, 0, 1/3, 2/3, 1, 1, 1, 1}}, 3, x]*)
    D[BSplineBasis[{3, knots}, 2, x], {x, 2}]
    (*9/2 (6 BSplineBasis[{1, {0, 0, 0, 0, 1/3, 2/3, 1, 1, 1, 1}}, 2, x] - 
           3 BSplineBasis[{1, {0, 0, 0, 0, 1/3, 2/3, 1, 1, 1, 1}}, 3, x]) - 
      3 (3 BSplineBasis[{1, {0, 0, 0, 0, 1/3, 2/3, 1, 1, 1, 1}}, 3, x] - 
         3 BSplineBasis[{1, {0, 0, 0, 0, 1/3, 2/3, 1, 1, 1, 1}}, 4, x])*)