Let be the open unit disk in the complex plane and be the set of all analytic functions on . An analytic self-map of induces a linear composition operator defined by
Moreover, let be the differentiation operator. We consider
In this paper, the author studies the operator defined by
acting between the weighted Bergman spaces
where , is the normalized Lebesgue measure, as well as . Boundedness and compactness of such operators acting between different weighted Bergman spaces as well as between a weighted Bergman space and the space of all bounded analytic functions on are characterized.