Bézier derivative control polygon ($C^{(r)}=\frac{n!}{(n-r)!}{\sum }_{j=0}^{n-r}{\Delta }^r{P}_j{B}_j^{n-r}$)

For a degree-$n$ Bézier curve, the $r$th derivative has degree at most $n-r$ and control points $n!{\Delta }^r{P}_j/(n-r)!$. The first derivative follows by differentiating the Bernstein basis and shifting one summation index; iteration proves the general formula. A parameter interval of length $h$ multiplies these derivative controls by $h^{-r}$.

Two adjacent Bézier curves of degrees $n,m$ and interval lengths $h,k$ have a $C^1$ join precisely when their endpoint positions and their derivatives in the common parameter agree. Parallel endpoint polygon edges alone give tangent-direction agreement; their scaled vectors must be equal for parametric first-derivative continuity.