${}$
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 $F\left(x\,y\left(x\right)\right)\=0$.