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Viewing: https://en.wikipedia.org/wiki/Implicit_function

Implicit function

Contents
    Algebraic functions
Caveats
Implicit differentiation
Implicit function theorem
    Marginal rate of technical substitution
See also
References

Part of a series of articles about
Calculus
R(x1,…,xn)=0, where R is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is y as an implicit function of −1 ≤ x ≤ 1 and g is a function of g, is the unique function giving a solution of the equation

y. This solution can then be written as

g is an implicit definition. For some functions g(x) = 2x − 1, then −1</sup>(''y'') = {{sfrac|1|2}}(''y'' + 1)"}},"i":0}}]}'/>g(y) = ⁠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 y − xe = 0.

Algebraic functions

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An algebraic function is a function that satisfies a polynomial equation whose coefficients are themselves polynomials. For example, an algebraic function in one variable an(x)yn+an−1(x)yn−1+⋯+a0(x)=0,

where the coefficients i</sub>''(''x'')"}},"i":0}}]}'/>ai(x) are polynomial functions of f is a 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:

y=±1−x2.

But even without specifying this explicit solution, it is possible to refer to the implicit solution of the unit circle equation as y, the same is not in general true for quintic and higher degree equations, such as

f.

Caveats

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Not every equation 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 R(x, y) = 0 can also have other pathologies. For example, the equation 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.

Implicit differentiation

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F(x,y)=0 defines y as a function x2+y2=1 cannot be represented globally as the graph of a single function F and the point x is needed. Tangent approximations to surfaces and higher-dimensional manifolds given by an equation or system of equations can be found by similar methods.

Implicit function theorem

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(x, y) satisfying 2</sup> + ''y''<sup>2</sup> = 1"}},"i":0}}]}'/>x + y = 1. Around point y(x). (Unlike in many cases, here this function can be made explicit as 1</sub>(''x'') = {{sqrt|1 − ''x''<sup>2</sup>}}"}},"i":0}}]}'/>g1(x) = √1 − x.) No such function exists around point (a, b) be a pair of real numbers such that ⁠∂R/∂y⁠ ≠ 0, then f that is defined and differentiable in some neighbourhood of x in this neighbourhood.

The condition (a, b) is a regular point of the implicit curve of implicit equation R is a multivariable polynomial. The set of the values of the variables that satisfy this relation is called an implicit curve if R(x, y) = 0 is an indifference curve for the quantities x.

Marginal rate of technical substitution

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Similarly, sometimes the level set 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 x* of the choice vector

See also

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References

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