Concept: The thermodynamic identity defines temperature and pressure as partial derivatives

Prerequisite concepts

The derivative can be interpreted physically

While the constituents of a derivative (the $f$ and $x$ in $\frac{df}{dx}$) can have physical interpretations, the derivative itself can have a different physical interpretation. For example, $\frac{dx}{dt}$ is a velocity. FIXME: find a better example

There are experimental limits to how $small$ of a change can be measured Partial derivatives that do not have the same variable(s) held constant (at the same values?) are not the same derivative The coefficients in a differentials equation are partial derivatives

First Law and thermodynamic identity

The thermodynamic identity defines temperature and pressure as partial derivatives

[T = \left(\frac{\partial U}{\partial S}\right)_V] [p = -\left(\frac{\partial U}{\partial V}\right)_S]