I have a given general monic polynomial
$$f(x,a) = 1 +a_1x +a_2 x^2 + ... +a_nx^n$$
with $n$ (possibly complex) roots $z_i$ with $i=1,\cdots,n$. I need the derivatives w. r. t. the parameters $a_j$ meaning $\dfrac{\partial z_i}{\partial a_j}$. Any Ideas? Is there a theorem or something that might help me?
Thanks a lot!