Implicit function

of the unit circle defines y as an implicit function of x if −1 ≤ x ≤ 1, and y is restricted to nonnegative values.

Intuitively, an inverse function is obtained from g by interchanging the roles of the dependent and independent variables.

Example: The product log is an implicit function giving the solution for x of the equation y − xex = 0.

For example, the equation x = 0 does not imply a function f(x) giving solutions for y at all; it is a vertical line.

In order to avoid a problem like this, various constraints are frequently imposed on the allowable sorts of equations or on the domain.

The implicit function theorem provides a uniform way of handling these sorts of pathologies.

Instead, one can totally differentiate R(x, y) = 0 with respect to x and y and then solve the resulting linear equation for ⁠dy/dx⁠ to explicitly get the derivative in terms of x and y.

Even when it is possible to explicitly solve the original equation, the formula resulting from total differentiation is, in general, much simpler and easier to use.

Consider This equation is easy to solve for y, giving where the right side is the explicit form of the function y(x).

Alternatively, one can totally differentiate the original equation: Solving for ⁠dy/dx⁠ gives the same answer as obtained previously.

The above formula comes from using the generalized chain rule to obtain the total derivative — with respect to x — of both sides of R(x, y) = 0: hence which, when solved for ⁠dy/dx⁠, gives the expression above.

In a less technical language, implicit functions exist and can be differentiated, if the curve has a non-vertical tangent.

The solutions of differential equations generally appear expressed by an implicit function.

[3] In economics, when the level set R(x, y) = 0 is an indifference curve for the quantities x and y consumed of two goods, the absolute value of the implicit derivative ⁠dy/dx⁠ is interpreted as the marginal rate of substitution of the two goods: how much more of y one must receive in order to be indifferent to a loss of one unit of x.

In this case the absolute value of the implicit derivative ⁠dK/dL⁠ is interpreted as the marginal rate of technical substitution between the two factors of production: how much more capital the firm must use to produce the same amount of output with one less unit of labor.

The implicit function theorem guarantees that the first-order conditions of the optimization define an implicit function for each element of the optimal vector x* of the choice vector x.