Implicit Differentiation

Background

So far, you may only have differentiated functions written in the form $y=f\left(x\right)$. But some functions are better described by an equation involving $x$ and $y$. For example, $x^{2}+y^{2}=16$ describes the graph of a circle with center $\left(0,0\right)$ and radius 4, and is really the graph of two functions: $y=\pm {\sqrt {16-x^{2}}}$.

Sometimes, functions described by equations in $x$ and $y$ are too hard to solve for $y$, for example $x^{3}+y^{3}=6xy$. This equation really describes 3 different functions of x, whose graph is the curve:

We want to find derivatives of these functions without having to solve for $y$ explicitly. We can do this by implicit differentiation, where we take the derivative of both sides of our equation with respect to $x$, and do some algebra steps to solve for $y'$ (or ${\dfrac {dy}{dx}}$ if you prefer), keeping in mind that $y$ is a function of $x$ in the equation.