imported>JJMC89 bot III: Moving :Category:Theorems in analysis to :Category:Theorems in mathematical analysis per Wikipedia:Categories for discussion/Log/2025 April 12#Category:Theorems in analysis

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Moving Category:Theorems in analysis to Category:Theorems in mathematical analysis per Wikipedia:Categories for discussion/Log/2025 April 12#Category:Theorems in analysis

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{{Short description|Mathematical relation consisting of a multi-variable function equal to zero}}
{{Calculus |Differential}}

In [[mathematics]], an '''implicit equation''' is a [[relation (mathematics)|relation]] of the form <math>R(x_1, \dots, x_n) = 0,</math> where {{mvar|R}} is a [[function (mathematics)|function]] of several variables (often a [[polynomial]]). For example, the implicit equation of the [[unit circle]] is <math>x^2 + y^2 - 1 = 0.</math>

An '''implicit function''' is a [[function (mathematics)|function]] that is defined by an implicit equation, that relates one of the variables, considered as the [[value (mathematics)|value]] of the function, with the others considered as the [[argument of a function|argument]]s.<ref name=Chiang>{{cite book |last=Chiang |first=Alpha C. |author-link=Alpha Chiang |title=Fundamental Methods of Mathematical Economics |location=New York |publisher=McGraw-Hill |edition=Third |year=1984 |isbn=0-07-010813-7 |url=https://archive.org/details/fundamentalmetho0000chia_b4p1 |url-access=registration }}</ref>{{rp|204–206}} For example, the equation <math>x^2 + y^2 - 1 = 0</math> of the [[unit circle]] defines {{mvar|y}} as an implicit function of {{mvar|x}} if {{math|−1 ≤ ''x'' ≤ 1}}, and {{mvar|y}} is restricted to nonnegative values.

The [[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 function]]s that are [[continuously differentiable]].

==Examples==

===Inverse functions===
A common type of implicit function is an [[inverse function]]. Not all functions have a unique inverse function. If {{mvar|g}} is a function of {{mvar|x}} that has a unique inverse, then the inverse function of {{mvar|g}}, called {{math|''g''−1}}, is the unique function giving a [[solution (mathematics)|solution]] of the equation

:<math> y=g(x) </math>

for {{mvar|x}} in terms of {{mvar|y}}. This solution can then be written as

:<math> x = g^{-1}(y) \,.</math>

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

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

'''Example:''' The [[product log]] is an implicit function giving the solution for {{mvar|x}} of the equation {{math|1=''y'' − ''xe''''x'' = 0}}.

===Algebraic functions===
{{main|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 {{mvar|x}} gives a solution for {{mvar|y}} of an equation

:<math>a_n(x)y^n+a_{n-1}(x)y^{n-1}+\cdots+a_0(x)=0 \,,</math>

where the coefficients {{math|''ai''(''x'')}} are polynomial functions of {{mvar|x}}. This algebraic function can be written as the right side of the solution equation {{math|1=''y'' = ''f''(''x'')}}. Written like this, {{mvar|f}} is a [[multi-valued function|multi-valued]] implicit function.

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

:<math>x^2+y^2-1=0 \,. </math>

Solving for {{mvar|y}} gives an explicit solution:

:<math>y=\pm\sqrt{1-x^2} \,. </math>

But even without specifying this explicit solution, it is possible to refer to the implicit solution of the unit circle equation as {{math|1=''y'' = ''f''(''x'')}}, where {{mvar|f}} is the multi-valued implicit function.

While explicit solutions can be found for equations that are [[quadratic equations|quadratic]], [[cubic equation|cubic]], and [[quartic equation|quartic]] in {{mvar|y}}, the same is not in general true for [[quintic equation|quintic]] and higher degree equations, such as

:<math> y^5 + 2y^4 -7y^3 + 3y^2 -6y - x = 0 \,. </math>

Nevertheless, one can still refer to the implicit solution {{math|1=''y'' = ''f''(''x'')}} involving the multi-valued implicit function {{mvar|f}}.

==Caveats==
Not every equation {{math|1=''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 {{math|1=''x'' − ''C''(''y'') = 0}} where {{mvar|C}} is a [[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 {{mvar|x}}-axis and "cutting away" some unwanted function branches. Then an equation expressing {{mvar|y}} as an implicit function of the other variables can be written.

The defining equation {{math|1=''R''(''x'', ''y'') = 0}} can also have other pathologies. For example, the equation {{math|1=''x'' = 0}} does not imply a function {{math|''f''(''x'')}} giving solutions for {{mvar|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 [[function domain|domain]]. The [[implicit function theorem]] provides a uniform way of handling these sorts of pathologies.

==Implicit differentiation==
In [[calculus]], a method called '''implicit differentiation''' makes use of the [[chain rule]] to differentiate implicitly defined functions.

To differentiate an implicit function {{math|''y''(''x'')}}, defined by an equation {{math|1=''R''(''x'', ''y'') = 0}}, it is not generally possible to solve it explicitly for {{mvar|y}} and then differentiate. Instead, one can [[total differentiation|totally differentiate]] {{math|1=''R''(''x'', ''y'') = 0}} with respect to {{mvar|x}} and {{mvar|y}} and then solve the resulting linear equation for {{math|{{sfrac|''dy''|''dx''}}}} to explicitly get the derivative in terms of {{mvar|x}} and {{mvar|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.

===Examples===
==== Example 1 ====
Consider

:<math>y + x + 5 = 0 \,.</math>

This equation is easy to solve for {{mvar|y}}, giving

:<math>y = -x - 5 \,,</math>

where the right side is the explicit form of the function {{math|''y''(''x'')}}. Differentiation then gives {{math|1={{sfrac|''dy''|''dx''}} = −1}}.

Alternatively, one can totally differentiate the original equation:

:<math>\begin{align}
\frac{dy}{dx} + \frac{dx}{dx} + \frac{d}{dx}(5) &= 0 \, ; \\[6px]
\frac{dy}{dx} + 1 + 0 &= 0 \,.
\end{align}</math>

Solving for {{math|{{sfrac|''dy''|''dx''}}}} gives

:<math>\frac{dy}{dx} = -1 \,,</math>

the same answer as obtained previously.

==== Example 2 ====
An example of an implicit function for which implicit differentiation is easier than using explicit differentiation is the function {{math|''y''(''x'')}} defined by the equation

:<math> x^4 + 2y^2 = 8 \,.</math>

To differentiate this explicitly with respect to {{mvar|x}}, one has first to get

:<math>y(x) = \pm\sqrt{\frac{8 - x^4}{2}} \,,</math>

and then differentiate this function. This creates two derivatives: one for {{math|''y'' ≥ 0}} and another for {{math|''y'' < 0}}.

It is substantially easier to implicitly differentiate the original equation:

:<math>4x^3 + 4y\frac{dy}{dx} = 0 \,,</math>

giving

:<math>\frac{dy}{dx} = \frac{-4x^3}{4y} = -\frac{x^3}{y} \,.</math>

==== Example 3 ====
Often, it is difficult or impossible to solve explicitly for {{mvar|y}}, and implicit differentiation is the only feasible method of differentiation. An example is the equation

:<math>y^5-y=x \,.</math>

It is impossible to [[algebraic expression|algebraically express]] {{mvar|y}} explicitly as a function of {{mvar|x}}, and therefore one cannot find {{math|{{sfrac|''dy''|''dx''}}}} by explicit differentiation. Using the implicit method, {{math|{{sfrac|''dy''|''dx''}}}} can be obtained by differentiating the equation to obtain

:<math>5y^4\frac{dy}{dx} - \frac{dy}{dx} = \frac{dx}{dx} \,,</math>

where {{math|1={{sfrac|''dx''|''dx''}} = 1}}. Factoring out {{math|{{sfrac|''dy''|''dx''}}}} shows that

:<math>\left(5y^4 - 1\right)\frac{dy}{dx} = 1 \,,</math>

which yields the result

:<math>\frac{dy}{dx}=\frac{1}{5y^4-1} \,,</math>

which is defined for

:<math>y \ne \pm\frac{1}{\sqrt[4]{5}} \quad \text{and} \quad y \ne \pm \frac{i}{\sqrt[4]{5}} \,.</math>

===General formula for derivative of implicit function===
If {{math|1=''R''(''x'', ''y'') = 0}}, the derivative of the implicit function {{math|''y''(''x'')}} is given by<ref name="Stewart1998">{{cite book | last = Stewart | first = James | title = Calculus Concepts And Contexts | publisher = Brooks/Cole Publishing Company | year = 1998 | isbn = 0-534-34330-9 | url-access = registration | url = https://archive.org/details/calculusconcepts00stew }}</ref>{{rp|§11.5}}

:<math>\frac{dy}{dx} = -\frac{\,\frac{\partial R}{\partial x}\,}{\frac{\partial R}{\partial y}} = -\frac {R_x}{R_y} \,,</math>

where {{math|''Rx''}} and {{math|''Ry''}} indicate the [[partial derivative]]s of {{mvar|R}} with respect to {{mvar|x}} and {{mvar|y}}.

The above formula comes from using the [[Chain rule#Multivariable case|generalized chain rule]] to obtain the [[total derivative]] — with respect to {{mvar|x}} — of both sides of {{math|1=''R''(''x'', ''y'') = 0}}:

:<math>\frac{\partial R}{\partial x} \frac{dx}{dx} + \frac{\partial R}{\partial y} \frac{dy}{dx} = 0 \,,</math>

hence

:<math>\frac{\partial R}{\partial x} + \frac{\partial R}{\partial y} \frac{dy}{dx} =0 \,,</math>

which, when solved for {{math|{{sfrac|''dy''|''dx''}}}}, gives the expression above.

==Implicit function theorem==
[[Image:Implicit circle.svg|thumb|right|200px|The unit circle can be defined implicitly as the set of points {{math|(''x'', ''y'')}} satisfying {{math|1=''x''2 + ''y''2 = 1}}. Around point {{mvar|A}}, {{mvar|y}} can be expressed as an implicit function {{math|''y''(''x'')}}. (Unlike in many cases, here this function can be made explicit as {{math|1=''g''1(''x'') = {{sqrt|1 − ''x''2}}}}.) No such function exists around point {{mvar|B}}, where the [[tangent space]] is vertical.]]
{{main|Implicit function theorem}}

Let {{math|''R''(''x'', ''y'')}} be a [[differentiable function]] of two variables, and {{math|(''a'', ''b'')}} be a pair of [[real number]]s such that {{math|1=''R''(''a'', ''b'') = 0}}. If {{math|{{sfrac|∂''R''|∂''y''}} ≠ 0}}, then {{math|1=''R''(''x'', ''y'') = 0}} defines an implicit function that is differentiable in some small enough [[neighbourhood (mathematics)|neighbourhood]] of {{open-open|''a'', ''b''}}; in other words, there is a differentiable function {{mvar|f}} that is defined and differentiable in some neighbourhood of {{mvar|a}}, such that {{math|1=''R''(''x'', ''f''(''x'')) = 0}} for {{mvar|x}} in this neighbourhood.

The condition {{math|{{sfrac|∂''R''|∂''y''}} ≠ 0}} means that {{math|(''a'', ''b'')}} is a [[singular point of a curve|regular point]] of the [[implicit curve]] of implicit equation {{math|1=''R''(''x'', ''y'') = 0}} where the [[tangent]] is not vertical.

In a less technical language, implicit functions exist and can be differentiated, if the curve has a non-vertical tangent.<ref name="Stewart1998"/>{{rp|§11.5}}

==In algebraic geometry==
Consider a [[relation (mathematics)|relation]] of the form {{math|1=''R''(''x''1, …, ''x''''n'') = 0}}, where {{mvar|R}} is a multivariable polynomial. The set of the values of the variables that satisfy this relation is called an [[implicit curve]] if {{math|1=''n'' = 2}} and an '''[[implicit surface]]''' if {{math|1=''n'' = 3}}. The implicit equations are the basis of [[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 set]]s.

==In differential equations==
The solutions of differential equations generally appear expressed by an implicit function.<ref>{{cite book |last=Kaplan |first=Wilfred |title=Advanced Calculus |location=Boston |publisher=Addison-Wesley |year=2003 |isbn=0-201-79937-5 }}</ref>

==Applications in economics==

===Marginal rate of substitution===

In [[economics]], when the level set {{math|1=''R''(''x'', ''y'') = 0}} is an [[indifference curve]] for the quantities {{mvar|x}} and {{mvar|y}} consumed of two goods, the absolute value of the implicit derivative {{math|{{sfrac|''dy''|''dx''}}}} is interpreted as the [[marginal rate of substitution]] of the two goods: how much more of {{mvar|y}} one must receive in order to be indifferent to a loss of one unit of {{mvar|x}}.

===Marginal rate of technical substitution===

Similarly, sometimes the level set {{math|''R''(''L'', ''K'')}} is an [[isoquant]] showing various combinations of utilized quantities {{mvar|L}} of labor and {{mvar|K}} of [[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 {{math|{{sfrac|''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.

===Optimization===
{{Main|Mathematical economics#Mathematical optimization}}

Often in [[economic theory]], some function such as a [[utility function]] or a [[Profit (economics)|profit]] function is to be maximized with respect to a choice vector {{mvar|x}} even though the objective function has not been restricted to any specific functional form. The [[implicit function theorem]] guarantees that the [[first-order condition]]s of the optimization define an implicit function for each element of the optimal vector {{math|''x''*}} of the choice vector {{mvar|x}}. When profit is being maximized, typically the resulting implicit functions are the [[labor demand]] function and the [[supply function]]s of various goods. When utility is being maximized, typically the resulting implicit functions are the [[labor supply]] function and the [[demand function]]s for various goods.

Moreover, the influence of the problem's [[Parameter#Mathematical functions|parameters]] on {{math|''x''*}} — the partial derivatives of the implicit function — can be expressed as [[total derivative]]s of the system of first-order conditions found using [[Differential of a function#Differentials in several variables|total differentiation]].
{{clear}}

==See also==
{{Div col|colwidth=20em}}
*[[Implicit curve]]
*[[Functional equation]]
*[[Level set]]
**[[Contour line]]
**[[Isosurface]]
*[[Marginal rate of substitution]]
*[[Implicit function theorem]]
*[[Logarithmic differentiation]]
*[[Polygonizer]]
*[[Related rates]]
*[[Folium of Descartes]]
{{Div col end}}

==References==
{{Reflist}}

==Further reading==
*{{cite book |first=K. G. |last=Binmore |author-link=Kenneth Binmore |chapter=Implicit Functions |title=Calculus |location=New York |publisher=Cambridge University Press |year=1983 |isbn=0-521-28952-1 |pages=198–211 |chapter-url=https://books.google.com/books?id=K8RfQgAACAAJ&pg=PA198 }}
*{{cite book |last=Rudin |first=Walter |author-link=Walter Rudin |title=Principles of Mathematical Analysis |url=https://archive.org/details/principlesofmath00rudi |url-access=registration |location=Boston |publisher=[[McGraw-Hill]] |year=1976 |isbn=0-07-054235-X |pages=[https://archive.org/details/principlesofmath00rudi/page/223 223–228] }}
*{{cite book |last=Simon |first=Carl P. |last2=Blume |first2=Lawrence |author-link2=Lawrence E. Blume |chapter=Implicit Functions and Their Derivatives |title=Mathematics for Economists |location=New York |publisher=W. W. Norton |year=1994 |isbn=0-393-95733-0 |pages=334–371 |chapter-url=https://books.google.com/books?id=l2nWMwEACAAJ&pg=PA334 }}

==External links==
*Archived at [https://ghostarchive.org/varchive/youtube/20211212/qb40J4N1fa4 Ghostarchive]{{cbignore}} and the [https://web.archive.org/web/20170507005435/https://www.youtube.com/watch?v=qb40J4N1fa4 Wayback Machine]{{cbignore}}: {{cite web |title=Implicit Differentiation, What's Going on Here? |series=Essence of Calculus |work=3Blue1Brown |date=May 3, 2017 |url=https://www.youtube.com/watch?v=qb40J4N1fa4&list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr |via=[[YouTube]] }}{{cbignore}}

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imported>JJMC89 bot III: Moving :Category:Theorems in analysis to :Category:Theorems in mathematical analysis per Wikipedia:Categories for discussion/Log/2025 April 12#Category:Theorems in analysis