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. …