QUESTION $3$

Prove that the following equation is valid;

Hint: Start with $(\frac{\text{d}\text{e}\text{l}\text{H}}{\text{d}\text{e}\text{l}\text{T}}{)}_P$ and $(\frac{\text{d}\text{e}\text{l}\text{H}}{\text{d}\text{e}\text{l}\text{P}}{)}_T$ and use the fact that, in mixed second parual derivative, the order of differentiation does not matter.