We give an explicit formula for the logarithmic potential of the asymptotic zero-counting measure of the sequence {(d(n)/dz(n)) (R(z) expT(z))}(n=1)(infinity). Here, R(z) is a rational function with at least two poles, all of which are distinct, and T(z) is a polynomial. This is an extension of a recent measure-theoretic refinement of Polya's Shire theorem for rational functions.