  In previous paper, we studied deformations $A_{\alpha,\beta}$ of the second
Weyl algebra and computed their derivations. In the present paper, we identify
the semiclassical limits $\mathcal{A}_{\alpha,\beta}$ of these deformations and
compute their Poisson derivations. Our results show that the first Hochschild
cohomology group HH$^1(A_{\alpha,\beta})$ is isomorphic to the first Poisson
cohomology group HP$^1( \mathcal{A}_{\alpha,\beta}).$
