2-Local derivations on the W-algebra W(2,2)

The present paper is devoted to study 2-local derivations on W-algebra $W(2,2)$ which is an infinite-dimensional Lie algebras with some out derivations. We prove that all 2-local derivations on the W-algebra $W(2,2)$ are derivation. We also give a complete classification of the 2-local derivation on the so called thin Lie algebra and prove that it admits a lots of 2-local derivations which are not derivations.