Implicit function theorem

The implicit function theorem gives conditions under which an equation locally determines some variables as differentiable functions of others. Its hypotheses use a nonzero Jacobian determinant with respect to the variables being solved for.

Jacobian matrix: A matrix of first-order partial derivatives of a vector-valued function. The theorem tests the Jacobian with respect to the variables to be determined.
Differentiable function: A function whose linear approximation error becomes negligible relative to distance near each point. Differentiability is the regularity assumed in the classical theorem.
Lagrange multipliers: A method for finding constrained extrema by relating gradients of an objective and constraint functions. The theorem helps justify local dependence of constrained stationary points on parameters.
Inverse function theorem: A theorem guaranteeing a differentiable local inverse when a map’s derivative is invertible. It solves for all output variables, while the implicit theorem solves for a selected subset.
Invertible matrix: A square matrix that has a multiplicative inverse. Invertibility of the relevant Jacobian block supplies the theorem’s local solvability condition.
Partial derivative: The derivative of a multivariable function with respect to one variable while holding others fixed. The theorem’s Jacobian is assembled from partial derivatives.
Comparative statics: The study of how equilibrium outcomes change when a model’s parameters change. Implicit differentiation derives local comparative-static effects from equilibrium equations.
Constant rank theorem: A theorem giving local coordinate forms for differentiable maps of constant rank. It handles rank-deficient maps under a constant-rank assumption beyond the basic square-Jacobian condition.
Chain rule: A rule for differentiating a composite function. Differentiating the defining equation yields the derivative formula for the implicit function.
Determinant: A scalar associated with a square matrix that detects whether the matrix is invertible. A nonzero determinant expresses the theorem’s condition for solving locally for selected variables.

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