  We address the question of how phonons of various frequencies contribute to
the superconducting transition temperature $T_c$ of a multi-band superconductor
by considering the functional derivatives of $T_c$ with respect to various
intraband and interband electron-phonon coupling functions. A general scheme
for computing such functional derivatives is developed. The proofs are given
that the functional derivatives which are diagonal in band indices are linear
in phonon energy $\Omega$ at small $\Omega$, while the functional derivatives
which are off-diagonal in band indices diverge at $\Omega$ = 0 as 1/$\Omega$.
The case of a two-band model for MgB$_{2}$ is treated numerically.
