Using the notation in the book by Piegl, when I have to compute the derivative of a B-Spline curve, I write $\sum_{i=0}^n N_{i,p}P_i$ on the knot vector $$U=(u_0=u_1=\ldots=u_p,u_{p+1},\ldots,u_{m-p-1},u_{m-p}=u_{m-p+1}=\ldots=u_m).$$ Then applying the formula for the derivative of a B-Spline curve I define
$$U'=(u_1=u_2=\ldots=u_p,u_{p+1},\dots,u_{m-p-1},u_{m-p}=\ldots=u_{m-1})$$ and the sum is now from $i=0$ to $i=n-1$. My question is: the $N_{0,p-1}$ has the support $[u_0,u_p)$, but $u_0$ is not present in $U'$. How can I solve this problem? Thanks
You don't explain the notation very well, and I don't have Piegl's book. However, I think the problem is this: the support of the basis function $N_{0,p-1}$ is the interval bounded by the first $p+1$ entries in its knot sequence. In the case of your derivative curve, those first $p+1$ entries are named $u_1, \ldots , u_{p+1}$.
If you write the second knot sequence as $(v_0, v_1, \ldots,)$ it will be easier for you to see what's happening.