We present a Sagnac interferometer that uses a nonlinear adaptive medium for detection. The nonlinearity of the medium being characterized by a finite response time, the detection has a finite frequency bandwidth determining two operational regimes: at low variation rate of the rotation speed the interferometer operates in the slow-light regime, providing a direct measurement of the angular acceleration. For high variation rate of the rotation speed the amplitude of the alternating component of the Sagnac phase shift is instead detected, with the continuous and low-frequency components filtered out by the adaptive nonlinear process in the medium.