= B-spline differentiation formula
{c}
{title2=$N_{i,d}'=\frac d{t_{i+d}-t_i}N_{i,d-1}-\frac d{t_{i+d+1}-t_{i+1}}N_{i+1,d-1}$}
For the partition-of-unity normalization, the derivative of a degree-$d$ <B-spline> is the displayed difference of two overlapping degree-$(d-1)$ <B-splines>. A zero denominator denotes a term with collapsed support, assigned zero. For simple knots the divided-difference representation $N_{i,d}(t)=(t_{i+d+1}-t_i)[t_i,\ldots,t_{i+d+1}](\tau-t)_+^d$ has the required support, smoothness and integral normalization. Differentiate the truncated power and apply the last-step <divided difference> recursion to obtain the formula. Repeated-knot versions follow by confluent limits, with one-sided interpretation where necessary.
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