Partial Derivatives
When dealing with Multivariable Functions, it is hard to talk about regular derivations. Instead, we calculate the partial derivatives of the function according to a certain variable. Which is written as: $$f_x(x,y) = \frac{\partial f}{\partial x}$$
When calculating the partial derivative for a certain variable, the other variables are treated as constants.
Partial derivation is VERY different from implicit differentiation since here, y is an independent variable and not a function that is defined according to x. For second-order derivatives and higher, the order of derivation doesn’t matter due to Clairaut’s Theorem