Lie algebra of Derivations as a functor?

Question

To an associative algebra $A$ one can associate a Lie algebra $\operatorname{Der} A$ of all derivations $D:A\to A$.

To any morphism of associative algebras $\alpha:A\to B$, how can one associate a morphism of Lie algebras $\alpha_\ast:\operatorname{Der} A\to \operatorname{Der} B$ or $\alpha^\ast:\operatorname{Der} B\to \operatorname{Der} A$ in a functorial way?