[Purpose/significance] During the evolution of public opinion, the derivation of public opinion could possess significant value for the forecasting and warning of public opinion both theoretically and empirically.[Method/process] To investigate the mechanism of public opinion derivation, this paper conducted the study using text clustering and DB cluster validity index. It proposed certain standards to judge the occurrence of public opinion derivation and its according velocity index. Furthermore, this paper used an well-known public opinion incident called "Classic Deadbeat" to conduct an empirical research.[Result/conclusion] The result of empirical study shows that:the derivative index reaches its climax during emergence phase and declined thereafter. The number of sub-topics reaches its climax during the integration phase and then declined thereafter; when the number of sub-topics decreased, the derivative index become negative,indicating that the public opinion become stabilized. When the public opinion incident reaches the disappearance phase, the number of sub topics become stable and the derivative index remain negative but approach zero. The study of derivative index of public opinion offers a new angle to study public opinion observation and prediction.