Derivable maps at commutative products on Banach algebras

Let A be a unital Banach algebra with unit e, M be a Banach A-bimodule, and \(w\in A\). In this paper, we characterize those continuous linear maps \(\delta :A\rightarrow M\) that satisfy one of the following conditions:

$$\begin{aligned} \delta (ab)= & {} \delta (a)b+a\delta (b), \\ 2\delta (w)= & {} \delta (a)b+a\delta (b),\\ \delta (ab)= & {} \delta (a)b+a\delta (b)-a\delta (e)b, \end{aligned}$$

for any \(a,b\in A\) with \(ab=ba=w\), where w is either a separating point with \(w\in Z(A)\) or an idempotent.