Let A be an algebra over K with multiplication $(x,y) \rightarrow x \cdot y$. A linear operator D on the vector space A is called a derivation of A if $D(x \cdot y)=(Dx) \cdot y + x \cdot (Dy)$ $( \forall x, y \in A)$.
Verify that the commutator $[ D,D' ]= D \circ D'-D' \cdot D $ is a derivation when D and D' are derivations of A.
So from definitions $[ D, D' ](x \cdot y)=(DD'-D'D)(x \cdot y)=DD'(x) \cdot y - D'D(x) \cdot y + x \cdot DD'(y) - x \cdot D'D(y)$.
This is what I think you have to do.