# Dynamic frontogenesis derivation

On page 142 of Lackmann's Midlatitude Synoptic Meteorology, Equation 6.7 gives the derivative of the Boussinesq buoyancy $$b = \frac{g \theta'}{\theta_0 (z)}$$ (perturbation potential temperature $$\theta'$$ and base state $$\theta_0 (z)$$) as

$$$$\frac{db}{dt} = -w \frac{g}{\theta_{00}} \frac{d \theta_0}{dz}$$$$

where $$\theta_{00}$$ is the base state potential temperature $$\theta_0 (z)$$ at $$z = 0$$. While I understand this to be an application of the chain rule

$$$$\frac{db}{dt} = \frac{\partial z}{\partial t} \frac{\partial \theta_0}{\partial z} \frac{\partial b}{\partial \theta_0},$$$$

I am unable to see why

$$$$\frac{\partial b}{\partial \theta_0} = - \frac{g}{\theta_{00}}$$$$

$$$$\frac{\partial b}{\partial \theta_0} = - \frac{g \theta'}{\theta_0^2 (z)}.$$$$