= Solution
Applying the preceding <derivative> identity repeatedly yields the <Bézier derivative control polygon> for every order:
$$
\boxed{P^{(r)}(t)=\frac{n!}{(n-r)!}\sum_{j=0}^{n-r}\Delta^rP_jB_j^{n-r}(t),\qquad0\le r\le n,}
$$
where the forward <finite difference> is $\Delta P_j=P_{j+1}-P_j$ and
$$
\Delta^rP_j=\sum_{k=0}^r(-1)^{r-k}\binom rkP_{j+k}.
$$
The induction step multiplies by the remaining degree $n-r$ and takes one more <finite difference>. <Derivatives> above order $n$ vanish. In particular,
$$
P^{(r)}(0)=\frac{n!}{(n-r)!}\Delta^rP_0,\qquad P^{(r)}(1)=\frac{n!}{(n-r)!}\Delta^rP_{n-r}.
$$
If the actual parameter interval has length $h$ and $t$ is its normalized coordinate, <derivatives> in the original parameter acquire the additional factor $h^{-r}$. These formulas allow <derivative> evaluation and join tests using only the <control polygon>.
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