Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)In this paper we introduce and study new concept of differentiability for fuzzy-set-valued functions. This derivative considers possible local interactivity in the process studied. Several properties of differentiability and integrability are investigated for the new concept and they are compared to similar fuzzy differentiabilities like Hukuhara differentiability and generalized Hukuhara differentiability. Furthermore, we establish theorems as the fundamental theorem of calculus. Ultimately, we exhibit some results for fuzzy initial value problem and an application. (C) 2016 Elsevier B.V. All rights reserved.3096480CNPq [305862/2013-8, 141085/2014-2]Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)