Differentiation Implicit

Talk about functions vs relations, and how relations are the more general case that we may actually wish to study in applications, but functions are easier to work wtih.

We know how to find slopes of curves that look like $y=f(x)$: we just take the derivative! But what if our curve is not given by an equation of this form? How can we find the change $dx$ and $dy$ in a situation like this? Example from Descartes:

We can differentiate using the chain rule and then solve for $dy/dx$! Indeed, this idea extends to be able to find the slope along a curve defining a relation at any point!

Examples: $$x^3+y^3=6xy$$ $$x^4+y^4=1$$ $$y\cos(x+y)=x$$