Monotonicity of polynomial critical points in their roots ($\frac{\partial \xi }{\partial a_j}=\frac{(\xi -a_j{)}^{-2}}{{\sum }_i(\xi -a_i{)}^{-2}}>0$)

For a monic polynomial $p(x)={\prod }_i(x-a_i)$ with distinct ordered real roots of a polynomial, each critical point $\xi$ lies between two consecutive roots by Rolle's theorem. Its logarithmic derivative gives ${\sum }_i(\xi -a_i{)}^{-1}=0$. Implicit differentiation gives the displayed formula, so moving any root to the right moves every ordered critical point to the right, while the roots remain distinct. Iterating this statement for derivatives preserves componentwise ordering of their roots. Weak inequalities extend by continuity to repeated roots.