We present for the first time the tangent linear method applied to the algorithm for solving integral equations of the theory of simple liquids. The purpose of this method is to compute the exact derivatives of the pair-correlation function g(r) with respect to density and temperature, which appear in thermodynamic properties and in the self-consistent procedure. An automatic differentiation tool is applied to the numerical algorithm of the integral equation method that combines the Newton-Raphson and the traditional iterative techniques. It results in a tangent linear code for the integral equation proposed by Vompe and Martynov, which is applied to calculate the thermodynamic properties and the bridge functions of the Lennard-Jones fluid. The results suggest that the tangent linear method should be preferred over the different strategies of differentiation that have been used up to now. Moreover, a local formulation of the Gibbs-Duhem relation is tractable in this framework. (C) 2001 American Institute of Physics.