Is there any simple way of computing the derivative to a complex polynomial of form
$$\left( z-z_{0} \right)\left( z-z_{1} \right)\left( z-z_{2} \right)...\left( z-z_{n} \right)=0$$
so that you do not have to multiply out the whole equation? I am trying to write a program which needs the exact derivative of the polynomial. My current approach is to write lots of complex buggy code to multiply out all powers of $z$, where I can then differentiate.