Thermodynamic Maxwell-like Relation in the Grand Canonical Ensemble

In the book "Physical Chemistry An Advanced Treatsie VIIIA" Eyring, Henderson and Jost use the thermodynamic relation:

$$\begin{equation} \left(\frac{\partial\rho}{\partial\mu}\right)=\rho\left(\frac{\partial\rho}{\partial P}\right) \end{equation}$$ $$\rho$$ is the number density ($$N/V$$), $$\mu$$ is the chemical potential and $$P$$ is the pressure. This relation is used in the grand canonical ensemble. I am having a hard time in deriving this relation. I have tried deriving the expression using Maxwell relations derived from the Landau grand potential, but I have not been successful so far. Any help in deriving this expression would be greatly appreciated.

Let me try: let $$\Omega(\mu,V,T)$$ be the grand-canonical potential, then the thermodynamic quantities conjugated to $$\mu$$ and $$V$$ (which are particle number $$N$$ and pressure $$P$$ respectively), are given by the relations $$N = \rho V = - \frac{\partial \Omega}{\partial \mu}; \;\;\;\; P = - \frac{\partial \Omega}{\partial V}.$$ Now we can write in a smart way $$\partial \rho/\partial \mu$$: $$\frac{\partial \rho}{\partial \mu} = \frac{\partial \rho}{\partial P}\frac{\partial P}{\partial \mu}.$$ Now the proof is complete once you are able to show that $$\partial P / \partial \mu = \rho$$, but this is very simple using the equations for $$N$$ and $$P$$ above: $$\frac{\partial P}{\partial \mu} = \frac{\partial }{\partial \mu} \left( - \frac{\partial \Omega}{\partial V} \right) = \frac{\partial }{\partial V} \left( - \frac{\partial \Omega}{\partial \mu} \right) = \frac{\partial }{\partial V} \left( \rho V \right) = \rho;$$ where I have simply swapped the derivatives.