Experimental results of prompt neutron decay constant α is useful information to validate numerical results of ω-eigenvalue for spatial and energetic fundamental mode. In order to accomplish the data assimilation technique using α, it is desirable to establish an efficient numerical calculation method for sensitivity coefficient analysis of α. For this purpose, the numerical calculation method using the first-order perturbation theory is investigated. A specific theoretical formula is derived to evaluate the sensitivity coefficient of α to nuclear data. The derived rigorous formula utilizes forward and adjoint eigenfunctions which consist of neutron flux and delayed neutron precursor densities. Using the prompt approximation, the derived formula can be simplified without the term involving the delayed neutron precursor densities. By calculating α using the multi-energy-group neutron transport code for an ICSBEP benchmark problem, the derived formula for sensitivity analysis using the perturbation theory is verified by comparing with the reference results using the direct method. Consequently, the efficient numerical procedures for uncertainty quantification of α can be established by the aid of the sensitivity coefficients based on the perturbation theory.