Question: How can one show the equivalence of two definitions of fluid isothermal compressibility?
I see the isothermal compressibility, $\beta_T$, commonly defined as: $$\tag{1} \beta_T=-\frac{1}{V_o}\left(\frac{\partial V}{\partial p}\right)_T$$
where $V_o$ is the initial or starting volume, $\partial V$ is the change in volume ($=V_o-V_{new}$), and $\partial p$ is the change in pressure ($=p_o-p_{new}$).
I have also seen the isothermal compressibility defined as: $$\tag{2} \beta_T=\frac{1}{\rho_o}\left(\frac{\partial \rho}{\partial p}\right)_T$$ where $\rho_o$ is the initial or starting density, $\partial \rho$ is the change in density($=\rho_o-\rho_{new}$), and $\partial p$ is the change in pressure ($=p_o-p_{new}$).
If I use the relationship between volume and density: $V=m/\rho$, where $m$ is the mass of the fluid, and I assume it to be constant, I suppose I can re-write Eqn (1) as:
$$\tag{3} \beta_T=-\rho_o\left(\frac{\partial \rho^{-1}}{\partial p}\right)_T$$
If I can do this, I am having trouble seeing how I can make my next step to arrive at Eqn (2).