How do I manipulate constraints in constraint derivatives of Legendre transformations?

Question

I having trouble understanding the constraint derivative in the highlighted part of the picture below, which allows to convert the one adimensional thermodynamical relation into its dimensional counterpart.
Specifically ,when doing the derivative of the product, done with $beta P = const$ ,why do the three derivates that follow  have different constraints: one is with $ P=const$, another one with $beta = const$ and another one with $beta P=const$. I would have written all of them with the same initial constraint: $beta P=const$, how do I realize how to change the constraint in each case?
For example for the case of the derivative with respect to $beta$  they write $ P=const$, but that seems wrong because when doing the derivative with respect to $beta $we are varying $beta$(because we're calculating how something varies when $beta$ varies) and if P is constant, then $beta P$ cannot be constant, which was the original constraint
Considering the following Legendre transformation:

From this, it follows that

SoterX · Answer

There is a misunderstanding in the meaning of leaving $P$ and $tilde P$ as a constant.
It is a general misunderstanding in the framework of thermodynamics, and a common mistake for many students.
When writing $left(frac{d mathcal G (beta, tilde P)}{d beta} right)_{tilde P = const.}$ you don' t mean that $tilde P$ is a constant. You mean that the partial derivative of function $mathcal G (beta, tilde P)$ is taken over the variable $beta$ AND that the two independent variables of $mathcal G$ are $beta $ and $tilde P$. So the subscript $tilde P = const.$ only tells you WHICH is the second variable.
The function $G(beta, P)$ is NOT a function of $tilde P$, but a function of $P$ instead. So, you need to perform a change of variables. You need to switch from the system of independent variables $(1) { beta , tilde P }$ to the system $(2) { beta, P }$.
Consider performing the transformation ${ alpha, tilde P } rightarrow { beta, P }$. The derivatives change becoming:
begin{equation}
left(frac{partial}{partial alpha}right)_{tilde P = const.} rightarrow left( frac{partial beta}{partial alpha} right)_{tilde P = const.} left(frac{partial}{partial beta}right)_{P = const.} + left(frac{partial P}{partial alpha}right)_{tilde P = const.} left(frac{partial}{partial P}right)_{beta = const.} 
end{equation}
and
begin{equation}
left(frac{partial}{partial tilde P}right)_{alpha= const.} rightarrow 
 left(frac{partial beta}{partial tilde P}right)_{alpha = const.} left(frac{partial}{partial beta}right)_{P= const.} + left(frac{partial P}{partial tilde P}right)_{alpha= const.} left(frac{partial}{partial P}right)_{beta = const.} .
end{equation}
In the specific case, the transformation is :
begin{align*}
&beta = alpha

&
P = tilde P / alpha,
end{align*}
and we are switching from $mathcal G (alpha, tilde P) = beta G(beta, P)$. Thus, we write:
begin{equation}
begin{split}
left(frac{d mathcal G (alpha, tilde P)}{d alpha} right)_{tilde P = const.}  = & left(frac{d beta G (beta, P)}{d alpha} right)_{tilde P = const.}

= & G(beta, P)
+& 
 beta left[  left( frac{partial beta}{partial alpha} right)_{tilde P = const.}  left(frac{d G (beta, P)}{d beta} right)_{P = const.} + left(frac{d P}{d alpha}right)_{tilde P = const.} left(frac{d G (beta, P)}{d P} right)_{beta = const.}   right].
end{split}
end{equation}
If you plug the transformation above in this relation you find the highlighted relation.

How do I manipulate constraints in constraint derivatives of Legendre transformations?

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