Prerequisite concepts
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
There are experimental limits to how $small$ of a change can be measured
The derivative can be interpreted physically
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Heat capacity lecture
Representations used
$df$
$\left(\frac{\partial f}{\partial x}\right)_y$