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**Implicit and explicit functions**

**In ****mathematics****, an implicit function is a ****function**** in which the ****dependent variable**** has not been given 'explicitly' in terms of the ****independent variable****. To give a function f explicitly is to provide a prescription for determining the output value of the function y in terms of the input value x:**

*y*=*f*(*x*).

**By contrast, the function is implicit if the value of y is obtained from x by solving an equation of the form:**

*R*(*x*,*y*) = 0.

**That is, it is defined as the ****level set**** of a function in two variables: one variable or the other may determine the other, but one is not given an explicit formula for one in terms of the other.**

**Implicit functions can often be useful in situations where it is inconvenient to solve explicitly an equation of the form R(x,y) = 0 for y in terms of x. Even if it is possible to rearrange this equation to obtain y as an explicit function f(x), it may not be desirable to do so since the expression of f may be much more complicated than the expression of R. In other situations, the equation R(x,y) = 0 may fail to define a function at all, and rather defines a kind of **

**multiple-valued function**

**. Nevertheless, in many situations, it is still possible to work with implicit functions. Some techniques from**

**calculus**

**, such as**

**differentiation**

**, can be performed with relative ease using**

*implicit differentiation*.**The ****implicit function theorem**** provides a link between implicit and explicit functions. It states that if the equation R(x, y) = 0 satisfies some mild conditions on its **

**partial derivatives**

**, then one can in principle solve this equation for**

*y*, at least over some small**interval**

**. Geometrically, the graph defined by**

*R*(*x*,*y*) = 0 will overlap**locally**

**with the graph of a function**

*y*=*f*(*x*).**Various ****numerical methods**** exist for solving the equation R(x,y)=0 to find an approximation to the implicit function y. Many of these methods are **

**iterative**

**in that they produce successively better approximations, so that a prescribed accuracy can be achieved. Many of these iterative methods are based on some form of**

**Newton's method**

**.**

Inverse functions

Inverse functions

**Implicit functions commonly arise as one way of describing the notion of an ****inverse function****. If f is a function, then the inverse function of f is a solution of the equation**

**for y in terms of x. Intuitively, an inverse function is obtained from f by interchanging the roles of the dependent and independent variables. Stated another way, the inverse function is the solution y of the equation**

*R*(*x*,*y*) =*x*−*f*(*y*) = 0.

**Examples.**

**The****natural logarithm***y*= ln(*x*) is the solution of the equation*x*ÿ−ÿ*e*^{y}ÿ=ÿ0.**The****product log****is an implicit function given by***x*ÿ−ÿ*y**e*^{y}ÿ=ÿ0.

### Algebraic functions

**Main article:**

**Algebraic function**

**An algebraic function is a solution y for an equation R(x,y) = 0 where R is a **

**polynomial**

**of two variables. Algebraic functions play an important role in**

**mathematical analysis**

**and**

**algebraic geometry**

**. A simple example of an algebraic function is given by the unit circle:**

*x*^{2}+*y*^{2}− 1 = 0.

**Solving for y gives**

**Note that there are two 'branches' to the implicit function: one where the sign is positive and the other where it is negative.**

## ÿCaveats

**Not every equation R(x,ÿy)ÿ=ÿ0 has a graph that is the graph of a function, the circle equation being one prominent example. Another example is an implicit function given by xÿ−ÿC(y)ÿ=ÿ0 where C is a **

**cubic polynomial**

**having a 'hump' in its graph. Thus, for an implicit function to be a true**

*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. A resulting formula may only then qualify as a legitimate explicit function.**The defining equation Rÿ=ÿ0 can also have other pathologies. For example, the implicit equation xÿ=ÿ0 does not define a function 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.**

## Implicit differentiation

**In ****calculus****, a method called implicit differentiation makes use of the ****chain rule**** to differentiate implicitly defined functions.**

**As explained in the introduction, y can be given as a function of x implicitly rather than explicitly. When we have an equation R(x,ÿy)ÿ=ÿ0, we may be able to solve it for y and then differentiate. However, sometimes it is simpler to differentiate R(x,ÿy) with respect to x and then solve forÿdy/dx.**

### ÿExamples

**1. Consider for example**

**This function normally can be manipulated by using ****algebra**** to change this ****equation**** to an ****explicit function****:**

**Differentiation then gives . Alternatively, one can differentiate the equation:**

**Solving for :**

**2. An example of an implicit function, for which implicit differentiation might be easier than attempting to use explicit differentiation, is**

**In order to differentiate this explicitly with respect to x, one would have to obtain (via algebra)**

**and then differentiate this function. This creates two derivatives: one for yÿ>ÿ0 and another forÿyÿ<ÿ0.**

**One might find it substantially easier to implicitly differentiate the implicit function;**

**thus,**

**3. Sometimes standard explicit differentiation cannot be used and, in order to obtain the derivative, another method such as implicit differentiation must be employed. An example of such a case is the implicit function y^{5}ÿ−ÿyÿ=ÿx. It is impossible to express y explicitly as a function of x and dy/dx therefore this cannot be found by explicit differentiation. Using the implicit method, dy/dx can be expressed:**

**factoring out shows that**

**which yields the final answer**

**where**

### Formula for two variables

**'The Implicit Function Theorem states that if F is defined on an open disk containing (a,b), where F(a,b) = 0, , and F_{x} and F_{y} are continuous on the disk, then the equation F(x,y) = 0 defines y as a function of x near the point (a,b) and the derivative of this function is given by...'^{[1]}^{:§ 11.5}**

*F*_{x},*F*_{y}indicates the derivative of*F*with respect to*x*and*y*

**The above formula comes from using the ****generalized chain rule**** to obtain the ****total derivative****—with respect to x—of both sides of F(x,ÿy)ÿ=ÿ0:**

### ÿMarginal rate of substitution

**Main article:**

**Marginal rate of substitution**

**In ****economics****, when the level set is an ****indifference curve****, the implicit derivative (or rather, −1 times the implicit derivative) is interpreted as the ****marginal rate of substitution**** of the two variables: how much more of y one must receive in order to be indifferent to a loss of 1 unit ofÿx.**

## ÿImplicit function theorem

**Main article:**

**Implicit function theorem**

**It can be shown that if R(x,y) is given by a **

**smooth**

**submanifold**

*M*in , and (*a*,*b*) is a point of this submanifold such that the**tangent space**

**there is not vertical (that is ), then**

*M*in some small enough**neighbourhood**

**of (**

*a*,*b*) is given by a**parametrization**

**(**

*x*,*f*(*x*)) where*f*is a**smooth function**

**. In less technical language, implicit functions exist and can be differentiated, unless the tangent to the supposed graph would be vertical. In the standard case where we are given an equation**

*F*(*x*,*y*) = 0

**the condition on F can be checked by means of **

**partial derivatives**

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http://en.wikipedia.org/wiki/Chain_rule

http://en.wikipedia.org/wiki/Chain_rule