Let $f$ be a multivariable function $f(x^{i}), i=(1,2,3....)$ its partial derivative is:
$ \left( \frac{\partial f}{\partial x^{i}} \right)\bigg|_{x^{j \ne i}=const}$
Can I define an equation where its left side implies $x^{j}$ constant and its right side implies a partial derivative of $x^{j}$?
For instance if $x^{j}$ is the temperature (T) and $x^{i}$ is the pressure (p) (where C is a constant and g is a temperature function) then if I say that the partial derivative of f with respect to temperature is a measurable property (U) being:
$ \left( \frac{\partial f}{\partial p} \right)\bigg|_{T} =U= C + \left( \frac{\partial g(T)}{\partial T} \right)$
Does this make sense? Because to measure U I should leave temperature constant but the practical case implies a change in temperature:
[Practical case: Heat of reaction and Van't Hoff realation
$r_{T,p} = \left( \frac{\partial H}{\partial \xi} \right)_{T, p} : r_{T,p} = -RT^{2}\frac{\partial }{\partial T}ln K(T,p)$
After integration
$ln\frac{K_2}{K_1}=-\frac{\Delta H}{R} \left( \frac{1}{T_2} - \frac{1}{T_1}\right) $ ]