Entropy is a fundamental concept in information theory but also in some AI algorithms, used for comparison of two distribution functions or measures that describe one specific event. When focusing on the differential (continuous) version of the entropy within the fuzzy measure framework, it is necessary to generalise all the essential concepts from the additive case to the fuzzy setup. This involves shifting from probability to fuzzy measures, from Lebesgue to Choquet integral and from Radon-Nikodym to Choquet-Radon-Nikodym derivatives. In the paper, two formulas for defining fuzzy entropy are proposed with the use of extended versions of the Choquet integral. Their basic properties are examined and compared with the additive case, and a relation with the fuzzy Kullback-Leibler divergence is derived. Using a novel insight into derivatives known as the resulting measure approach, some computations are presented to determine the final entropy value for two given measures.