Implicit function
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In mathematics, an implicit equation is a relation of the form where R is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is
An implicit function is a function that is defined by an implicit equation, that relates one of the variables, considered as the value of the function, with the others considered as the arguments.[1]: 204–206 For example, the equation of the unit circle defines y as an implicit function of x if −1 ≤ x ≤ 1, and y is restricted to nonnegative values.
The
Examples
Inverse functions
A common type of implicit function is an
for x in terms of y. This solution can then be written as
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
Intuitively, an inverse function is obtained from g by interchanging the roles of the dependent and independent variables.
Example: The
Algebraic functions
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
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
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:
Solving for y gives an explicit solution:
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
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
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
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 y(x), defined by an equation R(x, y) = 0, it is not generally possible to solve it explicitly for y and then differentiate. Instead, one can
Examples
Example 1
Consider
This equation is easy to solve for y, giving
where the right side is the explicit form of the function y(x). Differentiation then gives dy/dx = −1.
Alternatively, one can totally differentiate the original equation:
Solving for dy/dx gives
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 y(x) defined by the equation
To differentiate this explicitly with respect to x, one has first to get
and then differentiate this function. This creates two derivatives: one for y ≥ 0 and another for y < 0.
It is substantially easier to implicitly differentiate the original equation:
giving
Example 3
Often, it is difficult or impossible to solve explicitly for y, and implicit differentiation is the only feasible method of differentiation. An example is the equation
It is impossible to algebraically express y explicitly as a function of x, and therefore one cannot find dy/dx by explicit differentiation. Using the implicit method, dy/dx can be obtained by differentiating the equation to obtain
where dx/dx = 1. Factoring out dy/dx shows that
which yields the result
which is defined for
General formula for derivative of implicit function
If R(x, y) = 0, the derivative of the implicit function y(x) is given by[2]: §11.5
where Rx and Ry indicate the partial derivatives of R 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 R(x, y) = 0:
hence
which, when solved for dy/dx, gives the expression above.
Implicit function theorem
Let R(x, y) be a differentiable function of two variables, and (a, b) be a pair of real numbers 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 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 of the implicit curve of implicit equation 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.[2]: §11.5
In algebraic geometry
Consider a
In differential equations
The solutions of differential equations generally appear expressed by an implicit function.[3]
Applications in economics
Marginal rate of substitution
In economics, when the level set R(x, y) = 0 is an 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 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 showing various combinations of utilized quantities L of labor and 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 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
Often in
Moreover, the influence of the problem's parameters on x* — the partial derivatives of the implicit function — can be expressed as total derivatives of the system of first-order conditions found using total differentiation.
See also
References
- ISBN 0-07-010813-7.
- ^ ISBN 0-534-34330-9.
- ISBN 0-201-79937-5.
Further reading
- ISBN 0-521-28952-1.
- ISBN 0-07-054235-X.
- Simon, Carl P.; ISBN 0-393-95733-0.
External links
- Archived at Ghostarchive and the Wayback Machine: "Implicit Differentiation, What's Going on Here?". 3Blue1Brown. Essence of Calculus. May 3, 2017 – via YouTube.