The function whose rule is given by , is said to be defined explicitly. The function whose rule must be extracted from an equation of the form is said to be defined implicitly.
A simple example is the circle, defined by , where are two different explicit functions that can be extracted from the equation of the circle. The semicircle above the -axis is defined by ; and below, by .
Implicit differentiation is a technique by which can be obtained without necessarily having to solve for explicitly. It is merely the Chain rule applied to the identity .