Central extensions of some Lie algebras

We consider three Lie algebras: $Der \mathbb{C}((t))$, the Lie algebra of all derivations on the algebra $\mathbb{C}((t))$ of formal Laurent series; the Lie algebra of all differential operators on $\mathbb{C}((t))$; and the Lie algebra of all differential operators on $\mathbb{C}((t))\otimes \mathbb{C}^n.$ We prove that each of these Lie algebras has an essentially unique nontrivial central extension.