# Taking the derivative

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

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vela
Staff Emeritus
Homework Helper
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).