  We consider the variation of two fundamental types of zeta functions that
arise in the study of both physical and analytical problems in geometric
settings involving conical singularities. These are the Barnes zeta functions
and the Bessel zeta functions. Although the series used to define them do not
converge at zero, using methods of complex analysis we are able to calculate
the derivatives of these zeta functions at zero. These zeta functions depend
critically on a certain parameter, and we calculate the variation of these
derivatives with respect to the parameter. For integer values of the parameter,
we obtain a new expression for the variation of the Barnes zeta function with
respect to the parameter in terms of special functions. For the Bessel zeta
functions, we obtain two different expressions for the variation via two
independent methods. Of course, the expressions should be equal, and we verify
this by demonstrating several identities for both special and elementary
functions. We encountered these zeta functions while working with determinants
of Laplace operators on cones and angular sectors.
