Thermodynamic property relation derivation

Breeder18

1. The problem statement, all variables and given/known data
Derive an expression for the isentropic compressibility of a solid (BETA sub S) in terms of isothermal compressibility (beta sub T) and other properties normally tabulated.


2. Relevant equations
β_T≝-1/V (∂V/∂P)_T

β_S≝-1/V (∂V/∂P)_S


3. The attempt at a solution

I may be going in circles...
I realize some of my work, if not most, is redundant but I was hoping to see a possible method. Any and all help would be greatly appreciated.

P.S. I tried using the latex equation editor, but after 15 mins, wanted to shoot myself.

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