$$ \partial \rho = m / (\partial V) $$
$$ \partial \rho_k = m_k / (\partial V_k) $$
$$ \partial \rho_k = m_k / (\phi_k \partial V) $$
$$ \partial \rho = (\rho V) / (\partial V) $$
$$ \partial \rho_k = (\rho_k \phi_k V) / (\phi_k \partial V) $$
$$ \partial \rho_k = (\rho_k V) / (\partial V) $$
$$ \partial \rho = \rho (V / \partial V) $$
$$ \partial \rho_k = \rho_k (V / \partial V) $$
$$ \partial P = \sum_k \phi_k \partial P_k $$
$$ (\partial P/\partial \rho)\partial \rho = \sum_k \phi_k (\partial P_k/\partial \rho_k)\partial \rho_k $$
$$ (\partial P/\partial \rho)(\rho (V / \partial V)) = \sum_k \phi_k (\partial P_k/\partial \rho_k)(\rho_k (V / \partial V)) $$
$$ (\partial P/\partial \rho)\rho = \sum_k \phi_k (\partial P_k/\partial \rho_k)\rho_k $$