An implicit function is a function that is defined implicitly rather than explicitly. In other words, it is not given in the form \( y = f(x) \). Instead, an implicit function is defined by an equation that relates the variables \( x \) and \( y \) through an equation of the form \( F(x, y) = 0 \), where \( F \) is a function of both \( x \) and \( y \).
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An implicit function is a function specified through a relation between its input and output, rather than an explicit output formula. Locally, an equation may specify if one output is selected for each input in a suitable domain. Different branches can define different functions; the relation itself need not give a unique output globally. For example, gives the two branches on .
The implicit function theorem gives sufficient conditions for a unique local differentiable branch, while implicit differentiation computes its derivative by the chain rule. If is continuously differentiable and , that derivative is along the branch.