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, for example g(x, y) = c is a common representation of “implicit functions”, where g(x, y) is a function of both x and y, but we know y is still a function of x and x is an independent variable.

So how can we deal with such situations? Does the derivative even exist? The short answer is: yes, it does, but the way we evaluate it is slightly different. Consider the following example:

this here is an implicit function where y cannot be isolated entirely. In fact, this has the form g(y) = f(x), which means that in order to obtain the derivative y’(x) we need to do some clever manipulation. Firstly, we recall a simple fact from differentiation: if y = f(x), and we have a function g(y), then the derivative of g(y) with respect to x has to be equal to the derivative of y with respect to x times the derivative of g with respect to y. This is called the chain rule and is expressed as: