For helping to improve approximations to the density-functional exchange-correlation energy, E-xc[n], and its functional derivative, the difference between the second-order component of the correlation energy, E-c((2))[n], and the integral integral drv(c)((2))([n];r)n(r), involving its functional derivative, v(c)((2))([n];r), is given in terms of only the occupied Kohn-Sham orbitals and the exchange potential. The quantity 2E(c)((2))[n] is especially significant because it is the initial slope in the adiabatic connection formula for E-xc[n]. The analytic expression for 2E(c)((2))[n] -integral drv(c)((2))([n];r)n(r) is obtained for any spherically symmetric two-electron test density. Numerical examples are presented.