  This work presents an analysis of the functional derivative of the
superconducting transition temperature T$_c$ with respect to the
electron-phonon coupling function $\alpha^2F(\omega)$ [$\delta T_c/\delta
\alpha^2$F($\omega$)] and $\alpha^2F(\omega)$ spectrum of H$_3$S ($Im\bar3m$),
in the pressure range where the high-T$_c$ was measured (155-225 GPa). The
calculations are done in the framework of the Migdal-Eliashberg theory. We find
for this electron-phonon superconductor, a correlation between the maximums of
$\delta T_c/\delta \alpha^2$F($\omega$) and $\alpha^2F(\omega)$ with its higher
T$_c$. We corroborate this behavior in other electron-phonon superconductors by
analyzing data available in the literature, which suggests its validity in this
type of superconductors. The correlation observed could be considered as a
theoretical tool that in an electron-phonon superconductor, allows describing
qualitatively the proximity to its highest T$_c$, and determining the optimal
physical conditions (pressure, alloying or doping concentration) that lead to
the superconductor reaching its highest T$_c$ possible.

Superconductivity;Computational Physics