I'm studying ordinary derivative equations and I'm trying to understand how partial derivatives work. I'm trying to guess it using those equations here:
$$z = ax + by$$ $$z'= \frac{\partial z}{\partial x} = a + by'$$
and: $$y = zx$$ $$y'=\frac{\partial y}{\partial x} = z'x + x$$
I've looked other posts that asked similar things and searched the internet for information but I can only find explanations of partial derivative of a function ($\frac{\partial f}{\partial x}$)