Differentiation weights from nodal evaluations ($p^{\prime }(x_0)={\sum }_j{\ell }_j^{\prime }(x_0)p(a_j)$)

At distinct nodes $a_0,\dots ,a_n$, 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.