Implicit Function Theorem

When $\partial F/\partial y \neq 0$ , the level set $F(x,y)=0$ can be solved for $y$ as a function of $x$ , at least locally.

The Implicit Function Theorem guarantees that near a solution point of $F(x,y)=0$ where $\partial F/\partial y \neq 0$ , the level set is locally the graph of a function $y=g(x)$ , and gives its derivative without ever solving for $g$ explicitly: $g'(x) = -F_x/F_y$ .

Examples

$x^2+y^2=1$ near $(0,1)$ : $F_y=2y\neq0$ there, so the circle is locally $y=g(x)=\sqrt{1-x^2}$ .