There are many equations used to derive these four equations, but they have great physical significance. If you are a high school student, I suggest you recite these four formulas. If you are a college student, I can give you a detailed derivation process.
I just finished my sophomore year. . . I'm so sorry, teach fish to swim.
Definition of entropy
dS=dQ/T
first law of thermodynamics
dU=dQ+dW
Volume change work
dW=-pdV
therefore
dS=(dU+pdV)/T
That is dU=TdS-pdV.
Definition of enthalpy
H=U+pV
distinguish
dH=dU+pdV+Vdp
Bring dU=TdS-pdV and get
dH=TdS+Vdp
Definition of free energy
F=U-TS
difference
dF=dU-TdS-SdT
Bring it in with dU=TdS-pdV, and get
dF=-SdT-pdV
Definition of Gibbs function
G=U-TS+pV
The same difference
dG=dU-TdS-SdT+pdV+Vdp
Bring it in with dU=TdS-pdV, and get
dG=-SdT+VdP
This is the derivation of four formulas, which are generally used to derive Maxwell relations, that is, the relationship between the partial derivatives of four variables S, T, P and V. Maxwell relations are only used to push some messy things, and I have learned a little, so these formulas seem to me to be the time to prove some equations and various substitutions repeatedly, which is very annoying.