Implicit function theorem

If $F(x_0,y_0)=0$ and the partial derivative with respect to $y$ is invertible at $(x_0,y_0)$, then the nearby zero set is the graph $y=g(x)$ of a continuously differentiable function.

If a holomorphic function $F(z,w)$ satisfies $F(z_0,w_0)=0$ and $F_w(z_0,w_0)\ne 0$, then near $(z_0,w_0)$ its zero set is the graph $w=g(z)$ of a unique holomorphic function. The analogous statement with $F_z\ne 0$ uses $w$ as the local coordinate.

Implicit differentiation differentiates an identity $F(x,y(x))=0$ and uses the chain rule to solve for derivatives of the implicitly defined function. In one dimension, $y^{\prime }=-F_x/F_y$ when $F_y\ne 0$.