Implicit function

An implicit function is specified by an equation relating variables, rather than by an explicit formula isolating the dependent variable. Under suitable conditions, the equation determines that variable locally as a function of the others.

Implicit function theorem: A theorem giving conditions under which an equation locally determines one variable as a differentiable function of the others. It states when an implicit equation really defines a local function.
Explicit function: A function given by a formula that directly expresses its output in terms of its inputs. It contrasts with an implicit equation, which does not isolate the dependent variable.
Implicit differentiation: A method for finding derivatives by differentiating an equation that defines a function indirectly. It computes rates of change from the defining relation without first isolating the function.
Inverse function theorem: A theorem guaranteeing a local differentiable inverse when a function's derivative is invertible. It concerns inverting a map, while the implicit function theorem solves selected variables in an equation.
Jacobian matrix: A matrix of first-order partial derivatives that describes how a vector-valued function changes near a point. Its relevant determinant tests whether the equation can be solved locally for selected variables.
Partial derivative: A derivative measuring change in one input while other inputs are held fixed. Partial derivatives appear in the conditions and formulas for implicit differentiation.
Constraint manifold: A manifold consisting of points that satisfy one or more smooth constraint equations. Systems of implicit equations often describe the allowed states of a constrained model.
Global inverse theorem: A theorem giving conditions under which a locally invertible map is globally invertible. An implicit function may exist locally without extending as one global function.
Chain rule: A rule for differentiating a composite function by combining derivatives of its constituent functions. Differentiating an implicit relation uses the chain rule to connect dependent and independent variables.
Differentiability: The property of being well approximated near a point by a linear map. The standard implicit function theorem requires differentiability of the defining relation.

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