Strain rate is an important basic physical parameter in the fields of deformation observation, geodetic measurement, and geophysical monitoring. This paper proposes a novel fiber optic strain rate sensor (FOSRS) that can directly measure the strain rate through a differentiating interferometer that converts the strain rate to the optical phase. The sensing principle, sensitivity, resolution, and dynamic range of the proposed FOSRS are theoretically analyzed and verified by experiment. The experimental results show that the developed FOSRS with a 12.1 m sensing fiber has a flat sensitivity of 69.50 dB, a nanostrain rate ($\mathrm{n}\epsilon /\mathrm{s}$) resolution, and a dynamic range of better than 95 dB. An ultrahigh static resolution of 17.07 $\mathrm{p\epsilon }/\mathrm{s}$ can be achieved by using a 25.277 km sensing fiber for long baseline measurements. The proposed method significantly outperforms existing indirect measurement methods and has potential applications in geophysical monitoring and crustal deformation observation.