= Implicit function
{wiki}
An <implicit function> is a <function> specified through a relation between its input and output, rather than an explicit output formula. Locally, an <equation> $F(x,y)=0$ may specify $y=g(x)$ if one output $y$ is selected for each input $x$ in a suitable domain. Different branches can define different <functions>; the relation itself need not give a unique output globally. For example, $x^2+y^2=1$ gives the two branches $y=\pm\sqrt{1-x^2}$ on $-1<x<1$.
The <implicit function theorem> gives sufficient conditions for a unique local differentiable branch, while <implicit differentiation> computes its <derivative> by the <chain rule>. If $F$ is continuously differentiable and $F_y\ne0$, that <derivative> is $g'(x)=-F_x/F_y$ along the branch.
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