The Monte Carlo differential operator sampling method is applied to the computation of sensitivity coefficients of unresolved resonance probability table cross sections. Three new analytical benchmarks for verifying unresolved resonance treatments and sensitivity coefficient computations are developed. The method and its research code implementation are verified against these benchmarks, and agreement is observed. Numerical results for unresolved resonance sensitivity coefficients are obtained for the Big Ten benchmark and a simplified Molten Chloride Fast Reactor model. Energy-integrated eigenvalue sensitivity coefficients for the unresolved resonance range agree with MCNP6.2.0 calculations of these two models.