In this work we study optimization problems where both objective and constraints are given by fuzzy functions. In order to solve them, we first prove that such problems are equivalent to fuzzy optimization problems where the constraints are non-fuzzy (crisp) functions. Besides we prove a new and appropriate Karush-Kuhn-Tucker type necessary optimality condition based on the gH-differentiability and that has many computational advantages that we describe. The gH-derivative for fuzzy functions is a more general notion than Hukuhara and level-wise derivatives ones that are usually used in fuzzy optimization so far, in the sense that it can be applied to a wider number of fuzzy function classes than above concepts. With this new differentiability concept, we prove a KKT-type necessary optimality condition for fuzzy mathematical programming problems that is more operational and less restrictive that the few ones we can find in the literature so far.