A new kind of tangent cones is introduced, whose relationship to the contingent cone is discussed. With the introduced cones, a new kind of second-order tangent derivatives, termed  second-order composed tangent derivatives, is developed, and its relationship to other second-order composed tangent derivatives is discussed. Then, with the help of second-order composed tangent derivatives, optimality necessary conditions are established respectively for a Henig efficient solution and a globally proper efficient solution of set-valued optimization.