Often a relation (an expression in x and y), that has a graph but is not a function, needs to be analyzed. But the relation is not or cannot be solved for y. What to do? The answer is to use the technique of implicit differentiation. Assume there is a way to solve for y and differentiate using the Chain Rule. Whenever you get to the y,“differentiate” it by writing dy/dx. Then solve for dy/dx.
Here are several previous posts on this topic and how to go about using it.
Implicit differentiation of parametric equations These are BC topics
A Vector’s Derivative These are BC topics