The sub-harmonic resonance of Duffing oscillator with fractional-order derivative is studied, and the first-order approximate solution is obtained by the averaging method. The definitions of equivalent linear damping and equivalent linear stiffness for sub-harmonic resonance are presented, and the effects of the fractional-order parameters on the system dynamical characteristics are also analyzed. The amplitude-frequency equation for steady-state solution is established. The existence condition and stability condition for periodic response of sub-harmonic resonance are derived. At last the comparison between the analytical and numerical solution is fulfilled, and the result verifies the satisfactory precision of the approximate analytical solution. The numerical simulation is also used to analyze the effects of the parameters in fractional-order derivative on the existence condition, the stability condition and the amplitude-frequency curves of sub-harmonic resonance.