Derivatives of the Hurwitz Zeta function for rational arguments

The functional equation for the Hurwitz Zeta function ζ(s,a) is used to obtain formulas for derivatives of ζ(s,a) at negative odd s and rational a. For several of these rational arguments, closed-form expressions are given in terms of simpler transcendental functions, like the logarithm, the polygamma function, and the Riemann Zeta function.