  We study the extended prepotentials for the S-duality class of quiver gauge
theories, considering them as quasiclassical tau-functions, depending on gauge
theory condensates and bare couplings. The residue formulas for the third
derivatives of extended prepotentials are proven, which lead to effective way
of their computation, as expansion in the weak-coupling regime. We discuss also
the differential equations, following from the residue formulas, including the
WDVV equations, proven to be valid for the $SU(2)$ quiver gauge theories. As a
particular example we consider the constrained conformal quiver gauge theory,
corresponding to the Zamolodchikov conformal blocks by 4d/2d duality. In this
case part of the found differential equations turn into nontrivial relations
for the period matrices of hyperelliptic curves.
