# Equation of state in differential form

when studying the equation of states in astrophysics I have stumbled upon a differential form of it, i.e.

$$\frac{dP}{P}=\chi_T\frac{dT}{T}+\chi_P\frac{d\rho}{\rho}$$

where

$$\chi_T=\frac{T}{P}\frac{\partial P}{\partial T}\bigg|_{\rho,X_i}$$

$$\chi_\rho=\frac{\rho}{P}\frac{\partial P}{\partial \rho}\bigg|_{T,X_i}$$

$$\rho$$ - density

$$X_i$$ - an abundance of i's element

I would really use some hints on how to derive it with the help of the 1st law of thermodynamics.

You don't need the First Law of Thermodynamics at all. This is, in fact, just an application of basic multivariable calculus. The only assumption needed is that $$P$$ depends in some way on both $$T$$ and $$\rho$$, and that $$P$$ is differentiable with respect to $$T$$ and $$\rho$$.

So, if we have a function $$P(T,\rho)$$, we start by taking the differential:

$$dP=\frac{\partial P}{\partial T}\bigg|_{\rho,X_i} dT+\frac{\partial P}{\partial \rho}\bigg|_{T,X_i}d\rho$$

Then we divide by $$P$$:

$$\frac{dP}{P}=\frac{1}{P}\frac{\partial P}{\partial T}\bigg|_{\rho,X_i}dT+\frac{1}{P}\frac{\partial P}{\partial\rho}\bigg|_{T,X_i}d\rho$$

And then we multiply by $$1=\frac{T}{T}=\frac{\rho}{\rho}$$:

$$\frac{dP}{P}=\frac{T}{P}\frac{\partial P}{\partial T}\bigg|_{\rho,X_i}\frac{dT}{T}+\frac{\rho}{P}\frac{\partial P}{\partial\rho}\bigg|_{T,X_i}\frac{d\rho}{\rho}$$