Differentiation of Implicit Functions

So far, every function you've differentiated has been given explicitly — $y$ written directly as a formula in $x$, such as $y = f(x)$. But not every relation between $x$ and $y$ comes in that form. Consider $3x^2 + xy + y = 0$: here $y$ is tangled up with $x$ inside the same equation, though with some algebra it can still be solved for $y$ as a formula in $x$ in this particular case.

Many equations, however, resist being solved explicitly at all — a single equation in $x$ and $y$ can even trace out the graphs of several different functions joined together, each differentiable except where the pieces meet. A relation like this, where $y$ cannot conveniently be isolated as a standalone formula in $x$, defines $y$ as an implicit function of $x$. …