The main goal of this work is the further development of neoclassical analysis, which extends the scope and results of the classical mathematical analysis by applying fuzzy concepts to conventional mathematical objects, such as functions, sequences, and derivatives. This allows us to reflect and model vagueness and uncertainty of our knowledge, which results from imprecision of measurement and inaccuracy of computation. Basing on the theory of fuzzy limits, we construct a fuzzy extension for the classical theory of differentiation. The second part of this work, going after introduction, is devoted to the construction of fuzzy derivatives of real functions. Two kinds of fuzzy derivatives are introduced: weak and strong ones. Strong fuzzy derivatives are similar to ordinary derivatives of real functions, being their fuzzy generalizations. Weak fuzzy derivatives generate a new concept of a weak derivative even in a classical case when we assume complete precision.In the third part of this work, weak fuzzy derivatives and weak derivatives are applied to a study of monotonous functions. Different properties of such functions are obtained. Some of them are the same or at least similar to the properties of the differentiable functions while other properties differ in many aspects from those of the standard differentiable functions. Many classical results are obtained as direct corollaries of propositions for fuzzy derivatives, which are proved in this paper. Some of the classical results are extended and completed. The fifth part of this work contains several interpretations of fuzzy derivatives aiming at application of fuzzy differential calculus to solving practical problems. At the end, some open problems are formulated
