In this paper, we propose a time-varying modified minimum mean-squared error (MMSE) detector for the detection of higher data rate signals in a multirate asynchronous code-division multiple-access (CDMA) system which is signaled in a fast Rayleigh fading channel. The interference viewed by a higher data rate symbol will be periodic due to the presence of a lower data rate symbol which spans multiple higher data rate symbols. The detection is carried out on the basis of a modified MMSE criterion which incorporates differential detection and the ratio of channel coefficients in two consecutive observation intervals inherently compensating the fast variation of the channel due to fading. The numerical results obtained by the MMSE detector with time-varying detection show around 3 dB (M=2) and 6 dB (M=4) performance improvement at a BER of $10^{-3}$ in the AWGN channel, while introducing more computational complexity than the MMSE detector without time-varying detection. At a higher $E_b/N_0$, the proposed scheme can achieve a BER of approximately $10^{-3}$ in the presence of fast channel variation which is an improvement over other schemes.