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In mathematics, an implicit function is a generalization for the concept of a function in which the dependent variable may not be given explicitly in terms of the independent variable. To give a function f explicitly is to provide a prescription for calculating the output value of the function y in terms of the input value x of the function Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In experimental design, a dependent variable (also known as response variable, responding variable or regressand) is a factor whose values in different treatment conditions are compared. ... In an experimental design, the independent variable (argument of a function, also called a predictor variable) is the variable that is manipulated or selected by the experimenter to determine its relationship to an observed phenomenon (the dependent variable). ...

y = f(x)

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

R(x,y) = 0

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 functions given implicitly. Some techniques from calculus, such as differentiation, can be performed with relative ease using implicit differentiation. This diagram does not represent a true function, because the element 3 in X is associated with two elements, b and c, in Y. In mathematics, a multivalued function is a total relation; i. ... For other uses, see Calculus (disambiguation). ... Differentiation can mean the following: In biology: cellular differentiation; evolutionary differentiation; In mathematics: see: derivative In cosmogony: planetary differentiation Differentiation (geology); Differentiation (logic); Differentiation (marketing). ...

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). In multivariable calculus, a branch of mathematics, the implicit function theorem is a tool which allows relations to be converted to functions. ... In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In mathematics, a phenomenon is sometimes said to occur locally if, roughly speaking, it occurs on sufficiently small or arbitrarily small neighborhoods of points. ...

Examples

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 In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ...

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.

1. The natural logarithm y = ln(x) is the solution of the equation x - e^y = 0.
2. The product log is an implicit function given by x - y e^y = 0.

The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2. ... In mathematics, The Lambert W function, named after Johann Heinrich Lambert, also called the Omega function or product log, is the inverse function of f(w) = wew where ew is the natural exponential function and w is any complex number. ...

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: This article or section does not cite its references or sources. ... In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... Analysis has its beginnings in the rigorous formulation of calculus. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. ...

x² + y² − 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. Both branches are thought of belonging to the implicit function. In this way, implicit functions can be multiple-valued.

Caveats

Not every equation R(x,y) = 0 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 x - C(y) = 0 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. See also: cubic equation, Bézier curve, spline. ...

The defining equation R = 0 can also have other pathologies. For example, the implicit equation x = 0 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. In mathematics, the domain of a function is the set of all input values to the function. ...

Implicit differentiation

In calculus, a method called implicit differentiation can be applied to implicitly defined functions. This method is an application of the chain rule allowing one to calculate the derivative of a function given implicitly. For other uses, see Calculus (disambiguation). ... In calculus, the chain rule is a formula for the derivative of the composite of two 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,y) = 0, we may be able to solve it for y and then differentiate. However, sometimes it is simpler to differentiate R(x,y) with respect to x and then solve for dy / dx.

Examples

1. Consider for example

This function normally can be manipulated by using algebra to change this equation to an explicit function: This article is about the branch of mathematics. ... An equation is a mathematical statement, in symbols, that two things are the same (or equivalent). ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...

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, one would have to obtain (via algebra)

,

and then differentiate this function. This creates two derivatives: one for y > 0 and another for y < 0.

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 y³ − y = x. It is impossible to express y explicitly as a function of x (at least using elementary means, although the cubic formula will suffice for restricted values of x and y). Meaning, cannot be found by explicit differentiation. Using the implicit method, can be expressed: Graph of a cubic polynomial: y = x3/4 + 3x2/4 âˆ’ 3x/2 âˆ’ 2 = (1/4)(x + 4)(x + 1)(x âˆ’ 2) In mathematics, a cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power. ...

factoring out shows that

which yields the final answer

Formula for two variables

Suppose that y is bound to x by the equation F(x,y) = 0 and that y is a differentiable function of x. If F is differentiable, using the generalized chain rule on it yields In computer programming, a free variable is a variable referred to in a function that is not a local variable or an argument of that function. ... In calculus, the chain rule is a formula for the derivative of the composite of two functions. ...

.

Thus,

.

Implicit function theorem

Main article: Implicit function theorem

It can be shown that if R(x,y) is given by a smooth submanifold M in R², 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 In multivariable calculus, a branch of mathematics, the implicit function theorem is a tool which allows relations to be converted to functions. ... This is a glossary of terms specific to differential geometry and differential topology. ... The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ... See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ... In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ...

F(x,y) = 0

the condition on F can be checked by means of partial derivatives.^{[1]:§ 11.5} In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). ...

See also

• Level set
• Isosurface

In mathematics, a level set of a real-valued function f of n variables is a set of the form { (x1,...,xn) | f(x1,...,xn) = c } where c is a constant. ... Zirconocene with an isosurface showing areas of the molecule susceptible to electrophilic attack. ...

References

1. ^ Stewart, James (1998). Calculus Concepts And Contexts. Brooks/Cole Publishing Company. ISBN 0-534-34330-9. 

• Rudin, Walter (1976). Principles of Mathematical Analysis. McGraw-Hill. ISBN 0-07-054235-X. 
• Spivak, Michael (1965). Calculus on Manifolds. HarperCollins. ISBN 0-8053-9021-9. 
• Warner, Frank (1983). Foundations of Differentiable Manifolds and Lie Groups. Springer. ISBN 0-387-90894-3.