This paper deals with properties of pairs of functions, sets or multifunctions, that yield calculus rules for subdifferentials, normal cones and coderivatives, respectively, of operations involving these objects. The subdifferentials considered are introduced, in an axiomatic way, as operators, acting on lower semicontinuous functions defined on a class of normed spaces; the associated normal cones are defined as cones generated by subdifferentials of distance functions, and coderivatives are introduced, as usual, in terms of normal cones to graphs.