implicit differentiation

Implicit differentiation is a method of differentiation used if the equation cannot be easily transformed into a function (e.g. $y=\cdots$)

Steps for a univariate function ($x$ and $y$):

1. Differentiate both sides
2. Move $dy$ and $dx$ to separate sides
3. Cancel out extraneous $dy$ and $dx$‘s with algebra (divide both sides, factor, etc.)
4. Divide to get $\frac{dy}{dx}$ on one side

NOTE

in the differentiation step, treat a term that has different variables as a product of different functions with only one variable each and differentiate each function against its own variable. For example (using product rule):

$$\begin{array}{rl}xy& =1\\ x\cdot dy+y\cdot dx& =0\\ xdy& =-ydx\\ \frac{dy}{dx}& =-\frac{y}{x}\end{array}$$