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Sensitivity analysis is an important way for us to know how the input parameters will affect the output of a system. Therefore, recently, there is an increased interest in developing sensitivity analysis methods in continuous-energy Monte Carlo Code due to the fact that Monte Carlo method can perform high-fidelity simulations of nuclear reactor. Previous studies mainly focused on developing sensitivity analysis method suitable for analyze eigenvalue. There are relatively few researches for performing sensitivity analysis of generalized response function by using continuous-energy Monte Carlo code. So, in this work, the differential operator method (DOM) has been investigated and implemented in continuous-energy Reactor Monte Carlo code (RMC) to perform sensitivity analysis of generalized response function in the form of ratios of reaction rate. The DOM implemented in RMC is based on the analog Monte Carlo transport mode and non-analog Monte Carlo transport mode. The correctness of the newly implemented method has been verified by comparing the results with those calculated by using the collision history-based method through the Jezeble and Flattop benchmark problems. In general, the results given by the DOM agree well with those obtained by the collision history-based method with an accuracy of 5%. Moreover, it is also shown that the non-analog Monte Carlo transport mode can obtain lower relative standard deviation of the sensitivity coefficients than the analog Monte Carlo transport mode.