∀ {R : Type u_1} [inst : Semiring R] (f : Polynomial R),
  f.sylvesterDeriv =
    if hn : f.natDegree = 0 then 0
    else
      ((Polynomial.derivative f).sylvester f (f.natDegree - 1) f.natDegree).updateRow ⟨2 * f.natDegree - 2, ⋯⟩ fun j =>
        if ↑j = f.natDegree - 2 then 1 else if ↑j = 2 * f.natDegree - 2 then ↑f.natDegree else 0