Prerequisite concepts
Total differentials are linear
No matter how complicated a multivariable function is when written as an equation, the total differential of that function will be linear in the differential terms. In other words: $dF(x,y,z)=A dx + B dy + C dz.$
Representations used
Differentials allow the finding of partial derivatives when a variable cannot be solved for
There might be experimental limits on which quantities you can measure
You can flip a partial derivative if the same variable(s) are constant
Because total differentials are linear, one can find the total differential and then solve algebraically to obtain partial derivatives that might not be obtainable though differentiation.