How Do You Transition Between Derivative Equations in Fluid Dynamics?

[WesleyJA81]

In the text (attached) I can't figure out how they are making the jump from the first eqn to the second eqn. Any guidance would be helpful. Thanks

 

[vela]

Apparently, p=p₂ and ρ=ρ₂, and p is a function of 1/ρ. The quantities q, p₁, and ρ₁ are constants.

$$\frac{\gamma}{\gamma-1}\left(\frac{p}{\rho}-\frac{p_1}{\rho_1}\right)-\frac{1}{2}\left(\frac{1}{\rho_1}+\frac{1}{\rho}\right)(p-p_1)=q$$

If you let x=1/ρ, you can write the equation as

$$\frac{\gamma}{\gamma-1}\left(xp(x)-\frac{p_1}{\rho_1}\right)-\frac{1}{2}\left(\frac{1}{\rho_1}+x\right)(p(x)-p_1)=q$$

Differentiate that equation with respect to x and solve for p'(x).