Converting partial derivatives. Prove:

$(\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{T}}{)}_{V,n}=\frac{}{{}_T},$

given that ${}_T=-\frac{1}{V}(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{P}}{)}_{T,n},$ and $=\frac{1}{V}(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{T}}{)}_{P,n}.$

$2.$ Taking derivatives as a sanity check of the equation we just proved, $(\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{T}}{)}_{V,n}=\frac{}{{}_T},$ for an ideal gas.