Need help understanding this thermo. derivation

Im trying to understand this concept of deriving an expression for ([itex]\frac{\partial T}{\partial P}[/itex])_s in terms of T, V, C_p, [itex]\alpha[/itex], and [itex]\kappa[/itex]_T

([itex]\frac{\partial T}{\partial P}[/itex])_s is evaluated by measuring the temperature change and the specific volume change accompanying a small pressure change in a reversible adiabatic process.

I attached the derivation that the book does, but I cannot follow it. Any help would be greatly appreciated!

Thanks

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