The function $f$ whose rule is given by $f\left(x\right)\=x^2\+x\+1$, is said to be defined explicitly. The function $y\left(x\right)$ whose rule must be extracted from an equation of the form $F\left(x\,y\right)\=0$ is said to be defined implicitly.

A simple example is the circle, defined by $x^2\+y^2\=9$, where $y_{\pm }\left(x\right)\= \pm \sqrt{9-x^2}$ are two different explicit functions that can be extracted from the equation of the circle. The semicircle above the $x$-axis is defined by $y_{\+}\left(x\right)\=\sqrt{9-x^2}$; and below, by $y_{-}\left(x\right)\= -\sqrt{9-x^2}$.

Implicit differentiation is a technique by which $y\prime \left(x\right)$ can be obtained without necessarily having to solve for $y\left(x\right)$ explicitly. It is merely the Chain rule applied to the identity $F\left(x\,y\left(x\right)\right)\=0$.

If an equation of the form $F\left(x\,y\right)\=0$ defines a function $y\left(x\right)$ implicitly, then $F\left(x\,y\left(x\right)\right)\equiv 0$, that is, substitution of the expression $y\left(x\right)$ back into the equation results in an expression that is identically zero. This being the case, imagine that $y\left(x\right)$ has been found explicitly, so that $F\left(x\,y\left(x\right)\right)$ is identically zero.

Differentiate $F\left(x\,y\left(x\right)\right)$ with respect to $x$, and solve for $y\prime \left(x\right)$ in the resulting equation. Examples 2.5.1-3 illustrate exactly what this process entails.