  We prove formulas of different types that allow to calculate the Gerstenhaber
bracket on the Hochschild cohomology of an algebra using some arbitrary
projective bimodule resolution for it. Using one of these formulas, we give a
new short proof of the derived invariance of the Gerstenhaber algebra structure
on Hochschild cohomology. Also we give some new formulas for the Connes'
differential on the Hochschild homology that lead to formulas for BV
differential on the Hochschild cohomology in the case of symmetric algebras.
Finally, we use one of the obtained formulas to get a full description of the
BV structure and, correspondingly, the Gerstenhaber algebra structure on the
Hochschild cohomology of a class of symmetric algebras.
