Taking the derivative

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  • #1
WesleyJA81
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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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  • #2
vela
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Apparently, p=p2 and ρ=ρ2, and p is a function of 1/ρ. The quantities q, p1, and ρ1 are constants.

[tex]\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[/tex]

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

[tex]\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[/tex]

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

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