A new optical temporal first-order differentiator based on a double-clad and dual-core fiber structure is proposed. According to coupled-mode theory and vector analysis and through choosing reasonable structure parameters of the double-clad and dual-core fiber, a first-order optical temporal differentiator suitable to microwave regime is obtained and the processing signal bandwidth is 1.63 THz, the central frequency is 196.25 THz. The proposed differentiator has high accuracy and energy efficiency. In particular, the bandwidth and the central frequency of the differentiator can be tuned by changing the thickness of the clad to get the optical temporal differentiator with bandwidth and central frequency both tunable.