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A timedependent theory of a double quantum well resonant interband tunnel transistor is presented. The theory is based on a system of three linear differential equations for time-dependent resonant currents and charges stored in the wells. First the stationary resonant current-voltage curve is investigated. Then the general solution for the non-stationary tunnelling current is obtained by the perturbation method. The current is found to be an ultra-sensitive function of the timedependent small external voltage which is applied between the emitter and the second quantum well for detuning the resonances of quantum levels. Two nonstationary cases are investigated. For a step-function impulse of external voltage the characteristics of the response current are found. For a harmonic external voltage the frequency dependence of the response current is found. Matched and unmatched systems of barriers are investigated and found to exhibit significantly different behaviours.