  It is well-known that a formal deformation of a commutative algebra
${\mathcal A}$ leads to a Poisson bracket on ${\mathcal A}$ and that the
classical limit of a derivation on the deformation leads to a derivation on
${\mathcal A}$, which is Hamiltonian with respect to the Poisson bracket. In
this paper we present a generalisation of it for formal deformations of an
arbitrary noncommutative algebra ${\mathcal A}$. The deformation leads in this
case to a Poisson algebra structure on $\Pi({\mathcal A}):=Z({\mathcal
A})\times({\mathcal A}/Z({\mathcal A}))$ and to the structure of a
$\Pi({\mathcal A})$-Poisson module on ${\mathcal A}$. The limiting derivations
are then still derivations of ${\mathcal A}$, but with the Hamiltonian belong
to $\Pi({\mathcal A})$, rather than to ${\mathcal A}$. We illustrate our
construction with several cases of formal deformations, coming from known
quantum algebras, such as the ones associated with the nonabelian Volterra
chains, Kontsevich integrable map, the quantum plane and the quantised
Grassmann algebra.
