Let A be a Banach algebra, A*, A^** and A^*** be its first, second and third dual, respectively. Let R:A^***⟶A* be the restriction map, J: A*⟶A^*** be the canonical injection and Λ: A^***→A^*** be the composition of R and J. Let D:A⟶ A* be a continuous derivation and D^'':A^**⟶ A^*** be its second transpose. We obtain a necessary and sufficient condition for Λ∘D^'':A^**⟶ (A^**)* to be a derivation. We apply this to prove some results on weak amenability of second dual Banach algebras.

The first author was partly supported by a grant from IPM (No. 90430215).