Suppose that I have a polynomial $p(x) = c_0+c_1x^{-1}+\ldots c_{N-1} x^{-(N-1)}$ with roots $x_0,\ldots,x_{N-1}$, how do I derive an expression for the partial derivative of the root with respect to the coefficient i.e. $\frac{\partial x_i}{\partial c_k} \forall 0\leq i,k \leq N-1$
Thanks in advance
Let $x_0$ be a root. Then $$p(x_0)=0=c_0+c_1x_0^{-1}+\cdots+c_{N-1}x_0^{-(N-1)}$$
Clearly any given root is a function of the coefficients of the polynomial. Deriving both sides with respect to $c_k$ gives
$$0=x_0^{-k}-k c_k x_0^{-k-1}\frac{\partial x_0}{\partial c_k}-\sum_{j\neq k}jc_j x_0^{j-1}\frac{\partial x_0}{\partial c_k}$$ by the product rule and chain rule.
Thus \begin{align*} \frac{\partial x_0}{\partial c_k}&=\frac{x_0^{-k}}{\sum_{j} jc_jx_0^{-j-1}}\\ &=\frac{x_0}{\sum_{j}jc_jx_0^{k-j}} \end{align*}