Implicit Function Theorem

Definition

Two Variables Case

Suppose that $F(x,y)$ is a function with continuous partial derivatives $F_x$ and $F_y$ throughout an open region $R$ containing the point $(x_0,y_0)$. If for some constant $c$, $F(x_0,y_0)=c$ and $F_y(x_0,y_0)\ne 0$, then there exists a differentiable function $g$ that satisfies $y=g(x)$ defined in some Neighborhood of $x_0$ such that $F(x_0,g(x_0))=c$, and its derivative is given by

$\frac{dy}{dx}=-\frac{F_x}{F_y}$

Three Variables Case

Suppose that $F(x,y,z)$ is a function with continuous partial derivatives $F_x$, $F_y$, and $F_z$ throughout an open region $R$ containing the point $(x_0,y_0,z_0)$. If for some constant $c$, $F(x_0,y_0,z_0)=c$ and $F_z(x_0,y_0,z_0)\ne 0$, then there exists a differentiable function $g$ that satisfies $z=g(x,y)$ defined in some Neighborhood of $(x_0,y_0)$ such that $F(x_0,y_0,g(x_0,y_0))=c$, and its derivatives are given by

$\frac{\partial z}{\partial x}=-\frac{F_x}{F_z},\frac{\partial z}{\partial y}=-\frac{F_y}{F_z}$