  Let $\mathcal{A}$ be a unital Banach $*$-algebra and $\mathcal{M}$ be a
unital $*$-$\mathcal{A}$-bimodule. If $W$ is a left separating point of
$\mathcal{M}$, we show that every $*$-derivable mapping at $W$ is a Jordan
derivation, and every $*$-left derivable mapping at $W$ is a Jordan left
derivation under the condition $W \mathcal{A}=\mathcal{A}W$. Moreover we give a
complete description of linear mappings $\delta$ and $\tau$ from $\mathcal{A}$
into $\mathcal{M}$ satisfying $\delta(A)B^*+A\tau(B)^*=0$ for any $A, B\in
\mathcal{A}$ with $AB^*=0$ or $\delta(A)\circ B^*+A\circ\tau(B)^*=0$ for any
$A, B\in \mathcal{A}$ with $A\circ B^*=0$, where $A\circ B=AB+BA$ is the Jordan
product.
