Implicit differentiation in a nutshell

What is implicit differentiation? When we are dealing with derivatives of functions in calculus, we often encounter functions such as y = f(x) where some variable y can be explicitly expressed as a function of an independent variable x, so the differentiation process is straight-forward, the derivative of y is denoted as dy/dx or y’(x) and simply requires operating on f(x) using our standard rules of differentiation.

But not all functions can be written in this explicit form y = f(x). In fact, many times there are functions which cannot be separated that easily…