Let A be a unital Banach algebra with unit e, M be a Banach A-bimodule, and w ∈ A. In this paper, we characterize those continuous linear maps δ : A → M that satisfy one of the following conditions: δ(ab) = δ(a)b + aδ(b), 2δ(w) = δ(a)b + aδ(b), δ(ab) = δ(a)b + aδ(b) − aδ(e)b, for any a, b ∈ A with ab = ba = w,
