Sensitivity analysis is an important tool in mathematics as well as engineering with applications ranging from model analysis to optimal design to computational fluid dynamics. Several methodologies have been developed for the analysis and computation of sensitivities. This paper focuses on some theoretical aspects of using the Continuous Sensitivity Equation Method (CSEM) for design sensitivity analysis and computation. We concentrate on state equations which take the form of elliptic partial differential equations defined on parameter dependent domains. The PDEs are then formulated in an abstract form using elliptic operator equations. A framework is constructed which, under certain conditions, allows one to derive an operator sensitivity equation which can then be interpreted in the PDE sense.