Differentiation weights from nodal evaluations
= Differentiation weights from nodal evaluations
{title2=$p'(x_0)=\sum_j\ell_j'(x_0)p(a_j)$}
At distinct nodes $a_0,\ldots,a_n$, the <Lagrange interpolation polynomial> has basis $\ell_j(x)=\prod_{i\ne j}(x-a_i)/(a_j-a_i)$. Differentiating its exact expression for a polynomial of degree at most $n$ gives the displayed weights. Their uniqueness follows because the nodal evaluations form a basis of the <dual space> of that polynomial space.