Roots of the derivative as symmetric (?) functions of the roots of the polynomial

Question

Let $p(t)=(t^2-a_1^2)\ldots(t^2-a_n^2)$ be an even polynomial with distinct real non-zero roots. Can the roots of its derivative $p'(t)$ be expressed nicely (e.g. as rational symmetric functions) in terms of the roots of $p(t)$? How?