My assumption is that by (aG/aP), you are using the greek form of the derivative symbol. If this is correct, then the solution is as follows:
The partial derivative (dG/dP)_T, where G is the Gibbs free energy, P is pressure, and T is temperature, can be derived from the fundamental properties of Gibbs free energy. Gibbs free energy G is defined as G = U + PV - TS, where U is the internal energy, P is the pressure, V is the volume, T is the temperature, and S is the entropy.
The differential form of Gibbs free energy, expressed in terms of its natural variables T and P, is:
dG = -S dT + V dP
This expression indicates how G changes with small changes in temperature and pressure. From this differential form, we can directly extract the partial derivatives:
(dG/dT)_P = -S, which represents the negative of the entropy at constant pressure. (dG/dP)_T = V, which represents the volume at constant temperature.
Therefore, the equation for the partial derivative of Gibbs free energy with respect to pressure at constant temperature, (dG/dP)_T, is simply the volume V.
