The purpose of this research is toextendsome results introduced by Rockafellar[19] in finite-dimensioal spaces to general Banach space using the Housdoroff distance convergent instead of epigraphical convergent .These results are aplicationsto study the second-order epi-derivatives of function to classeand to study the second-order epi-derivatives of sum two convex functionand to studythe second-order epi-derivatives of Moreau-Yosida approximate function alsoto study ofthe second-order epi-derivatives of composition convex function with linear operator .