# Implicit function

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Mathematical relation consisting of a multi-variable function equal to zero

Part of a series of articles about Calculus ∫ a b f ′ ( t ) d t = f ( b ) − f ( a ) {\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)} Fundamental theorem Limits Continuity Rolle's theorem Mean value theorem Inverse function theorem Differential Definitions Derivative (generalizations) Differential infinitesimal of a function total Concepts Differentiation notation Second derivative Implicit differentiation Logarithmic differentiation Related rates Taylor's theorem Rules and identities Sum Product Chain Power Quotient L'Hôpital's rule Inverse General Leibniz Faà di Bruno's formula Reynolds Integral Lists of integrals Integral transform Leibniz integral rule Definitions Antiderivative Integral (improper) Riemann integral Lebesgue integration Contour integration Integral of inverse functions Nonelementary integral Integration by Parts Discs Cylindrical shells Substitution (trigonometric, tangent half-angle, Euler) Euler's formula Partial fractions (Heaviside's method) Changing order Reduction formulae Differentiating under the integral sign Risch algorithm Series Geometric (arithmetico-geometric) Harmonic Alternating Power Binomial Taylor Convergence tests Summand limit (term test) Ratio Root Integral Direct comparison Limit comparison Alternating series Cauchy condensation Dirichlet Abel Vector Gradient Divergence Curl Laplacian Directional derivative Identities Theorems Gradient Green's Stokes' Divergence Generalized Stokes Helmholtz decomposition Multivariable Formalisms Matrix Tensor Exterior Geometric Definitions Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian Hessian Theorems Clairaut's Fubini's Advanced Calculus on Euclidean space Generalized functions Limit of distributions Specialized Fractional Malliavin Stochastic Variations Miscellanea Precalculus History Glossary List of topics Integration Bee Mathematical analysis Nonstandard analysis v t e

In [mathematics](/source/Mathematics), an **implicit equation** is a [relation](/source/Relation_(mathematics)) of the form R ( x 1 , … , x n ) = 0 , {\displaystyle R(x_{1},\dots ,x_{n})=0,} where R is a [function](/source/Function_(mathematics)) of several variables (often a [polynomial](/source/Polynomial)). For example, the implicit equation of the [unit circle](/source/Unit_circle) is x 2 + y 2 − 1 = 0. {\displaystyle x^{2}+y^{2}-1=0.}

An **implicit function** is a [function](/source/Function_(mathematics)) that is defined by an implicit equation, that relates one of the variables, considered as the [value](/source/Value_(mathematics)) of the function, with the others considered as the [arguments](/source/Argument_of_a_function).[1]: 204–206 For example, the equation x 2 + y 2 − 1 = 0 {\displaystyle x^{2}+y^{2}-1=0} of the [unit circle](/source/Unit_circle) defines y as an implicit function of x, y = 1 − x 2 {\displaystyle y={\sqrt {1-x^{2}}}} , assuming −1 ≤ *x* ≤ 1 and y is restricted to nonnegative values. Some equations do not admit an [explicit solution](/source/Explicit_solution).

The [implicit function theorem](/source/Implicit_function_theorem) provides conditions under which some kinds of implicit equations define implicit functions, namely those that are obtained by equating to zero [multivariable functions](/source/Multivariable_function) that are [continuously differentiable](/source/Continuously_differentiable).

## Examples

### Inverse functions

A common type of implicit function is an [inverse function](/source/Inverse_function). Not all functions have a unique inverse function. If g is a function of x that has a unique inverse, then the inverse function of g, called *g*−1, is the unique function giving a [solution](/source/Solution_(mathematics)) of the equation

- y = g ( x ) {\displaystyle y=g(x)}

for x in terms of y. This solution can then be written as

- x = g − 1 ( y ) . {\displaystyle x=g^{-1}(y)\,.}

Defining *g*−1 as the inverse of g is an implicit definition. For some functions g, *g*−1(*y*) can be written out explicitly as a [closed-form expression](/source/Closed-form_expression) — for instance, if *g*(*x*) = 2*x* − 1, then *g*−1(*y*) = ⁠1/2⁠(*y* + 1). However, this is often not possible, or only by introducing a new notation (as in the [product log](/source/Product_log) example below).

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

**Example:** The [product log](/source/Product_log) is an implicit function giving the solution for x of the equation *y* − *xe**x* = 0.

### Algebraic functions

Main article: [Algebraic function](/source/Algebraic_function)

An **algebraic function** is a function that satisfies a polynomial equation whose coefficients are themselves polynomials. For example, an algebraic function in one variable x gives a solution for y of an equation

- a n ( x ) y n + a n − 1 ( x ) y n − 1 + ⋯ + a 0 ( x ) = 0 , {\displaystyle a_{n}(x)y^{n}+a_{n-1}(x)y^{n-1}+\cdots +a_{0}(x)=0\,,}

where the coefficients *ai*(*x*) are polynomial functions of x. This algebraic function can be written as the right side of the solution equation *y* = *f*(*x*). Written like this, f is a [multi-valued](/source/Multi-valued_function) implicit function.

Algebraic functions play an important role in [mathematical analysis](/source/Mathematical_analysis) and [algebraic geometry](/source/Algebraic_geometry). A simple example of an algebraic function is given by the left side of the unit circle equation:

- x 2 + y 2 − 1 = 0 . {\displaystyle x^{2}+y^{2}-1=0\,.}

Solving for y gives an explicit solution:

- y = ± 1 − x 2 . {\displaystyle y=\pm {\sqrt {1-x^{2}}}\,.}

But even without specifying this explicit solution, it is possible to refer to the implicit solution of the unit circle equation as *y* = *f*(*x*), where f is the multi-valued implicit function.

While explicit solutions can be found for equations that are [quadratic](/source/Quadratic_equations), [cubic](/source/Cubic_equation), and [quartic](/source/Quartic_equation) in y, the same is not in general true for [quintic](/source/Quintic_equation) and higher degree equations, such as

- y 5 + 2 y 4 − 7 y 3 + 3 y 2 − 6 y − x = 0 . {\displaystyle y^{5}+2y^{4}-7y^{3}+3y^{2}-6y-x=0\,.}

Nevertheless, one can still refer to the implicit solution *y* = *f*(*x*) involving the multi-valued implicit function f.

## Caveats

Not every equation *R*(*x*, *y*) = 0 implies a graph of a single-valued function, the circle equation being one prominent example. Another example is an implicit function given by *x* − *C*(*y*) = 0 where C is a [cubic polynomial](/source/Cubic_polynomial) having a "hump" in its graph. Thus, for an implicit function to be a *true* (single-valued) function it might be necessary to use just part of the graph. An implicit function can sometimes be successfully defined as a true function only after "zooming in" on some part of the x-axis and "cutting away" some unwanted function branches. Then an equation expressing y as an implicit function of the other variables can be written.

The defining equation *R*(*x*, *y*) = 0 can also have other pathologies. 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](/source/Function_domain). The [implicit function theorem](/source/Implicit_function_theorem) provides a uniform way of handling these sorts of pathologies.

## Implicit differentiation

This section is an excerpt from [Implicit differentiation](/source/Implicit_differentiation).[[edit](https://en.wikipedia.org/w/index.php?title=Implicit_differentiation&action=edit)]

In [calculus](/source/Calculus), [implicit differentiation](/source/Implicit_differentiation) is a method for finding the [derivative](/source/Derivative) of a [function](/source/Function_(mathematics)) that is defined by an equation rather than by an explicit formula. If an equation such as

- F ( x , y ) = 0 {\displaystyle F(x,y)=0}

defines y {\displaystyle y} as a function of x {\displaystyle x} , at least locally, implicit differentiation treats y {\displaystyle y} as a function y ( x ) {\displaystyle y(x)} and differentiates both sides of the equation with respect to x {\displaystyle x} . The method is an application of the [chain rule](/source/Chain_rule).[2]

Implicit differentiation is useful when solving explicitly for one variable is inconvenient, produces several [branches](/source/Branch_(complex_analysis)), or is not possible in [elementary](/source/Elementary_function) terms. For example, the circle x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} cannot be represented globally as the graph of a single function y = f ( x ) {\displaystyle y=f(x)} , but its upper and lower arcs can each be differentiated implicitly.

The method allows for the computation of the [tangent line approximation](/source/Tangent_line_approximation) to y ( x ) {\displaystyle y(x)} , given only the function F {\displaystyle F} and the point ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} at which the approximation is made, so that no other knowledge of the functional dependence of y {\displaystyle y} on x {\displaystyle x} is needed. Tangent approximations to [surfaces](/source/Surface_(mathematics)) and higher-dimensional [manifolds](/source/Manifold) given by an equation or system of equations can be found by similar methods.

## Implicit function theorem

Main article: [Implicit function theorem](/source/Implicit_function_theorem)

The unit circle can be defined implicitly as the set of points (*x*, *y*) satisfying *x*2 + *y*2 = 1. Around point A, y can be expressed as an implicit function *y*(*x*). (Unlike in many cases, here this function can be made explicit as *g*1(*x*) = √1 − *x*2.) No such function exists around point B, where the [tangent space](/source/Tangent_space) is vertical.

Let *R*(*x*, *y*) be a [differentiable function](/source/Differentiable_function) of two variables, and (*a*, *b*) be a pair of [real numbers](/source/Real_number) such that *R*(*a*, *b*) = 0. If ⁠∂*R*/∂*y*⁠ ≠ 0, then *R*(*x*, *y*) = 0 defines an implicit function that is differentiable in some small enough [neighbourhood](/source/Neighbourhood_(mathematics)) of (*a*, *b*); in other words, there is a differentiable function f that is defined and differentiable in some neighbourhood of a, such that *R*(*x*, *f*(*x*)) = 0 for x in this neighbourhood.

The condition ⁠∂*R*/∂*y*⁠ ≠ 0 means that (*a*, *b*) is a [regular point](/source/Singular_point_of_a_curve) of the [implicit curve](/source/Implicit_curve) of implicit equation *R*(*x*, *y*) = 0 where the [tangent](/source/Tangent) is not vertical.

In a less technical language, implicit functions exist and can be differentiated, if the curve has a non-vertical tangent.[3]: §11.5

## In algebraic geometry

Consider a [relation](/source/Relation_(mathematics)) of the form *R*(*x*1, …, *x**n*) = 0, where R is a multivariable polynomial. The set of the values of the variables that satisfy this relation is called an [implicit curve](/source/Implicit_curve) if *n* = 2 and an **[implicit surface](/source/Implicit_surface)** if *n* = 3. The implicit equations are the basis of [algebraic geometry](/source/Algebraic_geometry), whose basic subjects of study are the simultaneous solutions of several implicit equations whose left-hand sides are polynomials. These sets of simultaneous solutions are called [affine algebraic sets](/source/Affine_algebraic_set).

## In differential equations

The solutions of differential equations generally appear expressed by an implicit function.[4]

## Applications in economics

### Marginal rate of substitution

In [economics](/source/Economics), when the level set *R*(*x*, *y*) = 0 is an [indifference curve](/source/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](/source/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.

### Marginal rate of technical substitution

Similarly, sometimes the level set *R*(*L*, *K*) is an [isoquant](/source/Isoquant) showing various combinations of utilized quantities L of labor and K of [physical capital](/source/Physical_capital) each of which would result in the production of the same given quantity of output of some good. In this case the absolute value of the implicit derivative ⁠*dK*/*dL*⁠ is interpreted as the [marginal rate of technical substitution](/source/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.

### Optimization

Main article: [Mathematical economics § Mathematical optimization](/source/Mathematical_economics#Mathematical_optimization)

Often in [economic theory](/source/Economic_theory), some function such as a [utility function](/source/Utility_function) or a [profit](/source/Profit_(economics)) function is to be maximized with respect to a choice vector x even though the objective function has not been restricted to any specific functional form. The [implicit function theorem](/source/Implicit_function_theorem) guarantees that the [first-order conditions](/source/First-order_condition) of the optimization define an implicit function for each element of the optimal vector *x** of the choice vector x. When profit is being maximized, typically the resulting implicit functions are the [labor demand](/source/Labor_demand) function and the [supply functions](/source/Supply_function) of various goods. When utility is being maximized, typically the resulting implicit functions are the [labor supply](/source/Labor_supply) function and the [demand functions](/source/Demand_function) for various goods.

Moreover, the influence of the problem's [parameters](/source/Parameter#Mathematical_functions) on *x** — the partial derivatives of the implicit function — can be expressed as [total derivatives](/source/Total_derivative) of the system of first-order conditions found using [total differentiation](/source/Differential_of_a_function#Differentials_in_several_variables).

## See also

- [Implicit curve](/source/Implicit_curve)

- [Functional equation](/source/Functional_equation)

- [Level set](/source/Level_set) - [Contour line](/source/Contour_line) - [Isosurface](/source/Isosurface)

- [Marginal rate of substitution](/source/Marginal_rate_of_substitution)

- [Implicit function theorem](/source/Implicit_function_theorem)

- [Logarithmic differentiation](/source/Logarithmic_differentiation)

- [Polygonizer](/source/Polygonizer)

- [Related rates](/source/Related_rates)

- [Folium of Descartes](/source/Folium_of_Descartes)

## References

1. **[^](#cite_ref-Chiang_1-0)** [Chiang, Alpha C.](/source/Alpha_Chiang) (1984). [*Fundamental Methods of Mathematical Economics*](https://archive.org/details/fundamentalmetho0000chia_b4p1) (Third ed.). New York: McGraw-Hill. [ISBN](/source/ISBN_(identifier)) [0-07-010813-7](https://en.wikipedia.org/wiki/Special:BookSources/0-07-010813-7).

1. **[^](#cite_ref-Implicit_differentiation_OpenStax_2-0)** Strang, Gilbert; Herman, Edwin (2016). "3.8 Implicit Differentiation". [*Calculus Volume 1*](https://openstax.org/books/calculus-volume-1/pages/3-8-implicit-differentiation). OpenStax. [ISBN](/source/ISBN_(identifier)) [978-1-938168-02-4](https://en.wikipedia.org/wiki/Special:BookSources/978-1-938168-02-4).

1. **[^](#cite_ref-Stewart1998_3-0)** Stewart, James (1998). [*Calculus Concepts And Contexts*](https://archive.org/details/calculusconcepts00stew). Brooks/Cole Publishing Company. [ISBN](/source/ISBN_(identifier)) [0-534-34330-9](https://en.wikipedia.org/wiki/Special:BookSources/0-534-34330-9).

1. **[^](#cite_ref-4)** Kaplan, Wilfred (2003). *Advanced Calculus*. Boston: Addison-Wesley. [ISBN](/source/ISBN_(identifier)) [0-201-79937-5](https://en.wikipedia.org/wiki/Special:BookSources/0-201-79937-5).

## Further reading

- [Binmore, K. G.](/source/Kenneth_Binmore) (1983). ["Implicit Functions"](https://books.google.com/books?id=K8RfQgAACAAJ&pg=PA198). *Calculus*. New York: Cambridge University Press. pp. 198–211. [ISBN](/source/ISBN_(identifier)) [0-521-28952-1](https://en.wikipedia.org/wiki/Special:BookSources/0-521-28952-1).

- [Rudin, Walter](/source/Walter_Rudin) (1976). [*Principles of Mathematical Analysis*](https://archive.org/details/principlesofmath00rudi). Boston: [McGraw-Hill](/source/McGraw-Hill). pp. [223–228](https://archive.org/details/principlesofmath00rudi/page/223). [ISBN](/source/ISBN_(identifier)) [0-07-054235-X](https://en.wikipedia.org/wiki/Special:BookSources/0-07-054235-X).

- Simon, Carl P.; [Blume, Lawrence](/source/Lawrence_E._Blume) (1994). ["Implicit Functions and Their Derivatives"](https://books.google.com/books?id=l2nWMwEACAAJ&pg=PA334). *Mathematics for Economists*. New York: W. W. Norton. pp. 334–371. [ISBN](/source/ISBN_(identifier)) [0-393-95733-0](https://en.wikipedia.org/wiki/Special:BookSources/0-393-95733-0).

## External links

- Archived at [Ghostarchive](https://ghostarchive.org/varchive/youtube/20211212/qb40J4N1fa4) and the [Wayback Machine](https://web.archive.org/web/20170507005435/https://www.youtube.com/watch?v=qb40J4N1fa4): ["Implicit Differentiation, What's Going on Here?"](https://www.youtube.com/watch?v=qb40J4N1fa4&list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr). *3Blue1Brown*. Essence of Calculus. May 3, 2017 – via [YouTube](/source/YouTube).

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Adapted from the Wikipedia article [Implicit function](https://en.wikipedia.org/wiki/Implicit_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Implicit_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.