Let $\Cal A$ be a unital Banach $*$-algebra and $\Cal M$ be a unital $*$-$\Cal A$-bimodule. For $W$ a left separating point of $\Cal M$, the authors 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 $W\Cal A=\Cal A W$. They give a complete description of all linear mappings $\delta$ and $\tau$ from $\Cal A$ into $\Cal M$ satisfying $$ \delta(A)B^*=0 \quad\text{ or }\quad \delta(A)\circ B^*+A\circ\tau(B)^*=0 $$ for any $A,B\in \Cal A$ with $A\circ B^*=0$, where $A\circ B=AB+BA$ is the Jordan product. These new and interesting theorems can be regarded as an improvement and extension of previous results.