Differentiation of the Choquet integral of a nonnegative measurable function

Differentiation of the Choquet integral of a nonnegative measurable function taken with respect to a fuzzy measure on fuzzy measure space is proposed. First, the real interval limited Choquet integral is defined for a preparation, then the upper differential coefficient, the lower differential coefficient, the differential coefficient and the derived function of the Choquet integral along the range of an integrated function are defined by the limitation process of the interval limited Choquet integral. Two examples are given, where the nonnegative measurable functions are either a simple function or a triangular function. Basic properties of differentiation about sum and multiple with constant, addition, subtraction, multiplication and division are shown. It should be noted that the derived function of the Choquet integral of a composite function with sum of nonnegative measurable functions is not always equal to the sum of each derived functions of the Choquet integrals of these functions. Moreover, they are applied to the capital investment decision making problem.