NationMaster - Encyclopedia: Inverse functions and differentiation

In mathematics, the inverse of a function $y = f (x)$ is a function that, in some fashion, "undoes" the effect of $f$ (see inverse function for a formal and detailed definition). The inverse of $f$ is denoted $f - 1$ . The statements y=f(x) and x=f^-1(y) are equivalent. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Inter. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ...

The two derivatives are, as the Leibniz notation suggests, reciprocal, that is In calculus, the Leibniz notation, named in honor of the 17th century German philosopher and mathematician Gottfried Wilhelm Leibniz (pronounced LIBE nits) was originally the use of dx and dy and so forth to represent infinitely small increments of quantities x and y, just as Î”x and Î”y represent finite... In mathematics, the reciprocal, or multiplicative inverse, of a number x is the number which, when multiplied by x, yields 1. ...

$frac{dx}{dy},cdot, frac{dy}{dx} = 1.$

This is a direct consequence of the chain rule, since In calculus, the chain rule is a formula for the derivative of the composition of two functions. ...

$frac{dx}{dy},cdot, frac{dy}{dx} = frac{dx}{dx}$

and the derivative of $x$ with respect to $x$ is 1.

Writing explicitely the dependence of y on x and the point at which the differentaition takes place and using Lagrange's notation, the formula for the derivative of the inverse becomes

$left[f^{-1}right]'(a)=frac{1}{f'left[f^{-1}(a)right]}.$

Geometrically, a function and inverse function have graphs that are reflections, in the line y=x. This reflection operation turns the gradient of any line into its reciprocal. The word reflection (also spelt reflexion in British English) can refer to several different concepts: In mathematics, reflection is the transformation of a space. ... In the above two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows. ... In mathematics, the reciprocal, or multiplicative inverse, of a number x is the number which, when multiplied by x, yields 1. ...

Examples

$y = x 2$ (for positive $x$ ) has inverse $x = sqrt{y}$ .

$frac{dy}{dx} = 2x mbox{ }mbox{ }mbox{ }mbox{ }; mbox{ }mbox{ }mbox{ }mbox{ } frac{dx}{dy} = frac{1}{2sqrt{y}}$

$frac{dy}{dx},cdot,frac{dx}{dy} = 2x cdotfrac{1}{2sqrt{y}} = frac{2x}{2x} = 1.$

At x=0, however, there is a problem: the graph of the square root function becomes vertical, corresponding to a horizontal tangent for the square function.

$y = e x$ has inverse $x = ln(y)$ (for positive $y$ )

$frac{dy}{dx} = e^x mbox{ }mbox{ }mbox{ }mbox{ }; mbox{ }mbox{ }mbox{ }mbox{ } frac{dx}{dy} = frac{1}{y}$

$frac{dy}{dx},cdot,frac{dx}{dy} = e^x cdot frac{1}{y} = frac{e^x}{e^x} = 1$

Additional properties

Integrating this relationship gives

${f^{-1}}(y)=intfrac{1}{f'(x)},cdot,{dx} + c.$

This is only useful if the integral exists. In particular we need

f'(x)

to be non-zero across the range of integration.

It follows that functions with continuous derivative have inverses in a neighbourhood of every point where the derivative is non-zero. This need not be true if the derivative is not continuous.

In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ...

Related topics

calculus, inverse functions, chain rule, inverse function theorem, implicit function theorem. Calculus is a central branch of mathematics, developed from algebra and geometry, and built on two major complementary ideas. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... In calculus, the chain rule is a formula for the derivative of the composition of two functions. ... In mathematics, the inverse function theorem gives sufficient conditions for a vector-valued function to be invertible on an open region containing a point in its domain. ... In mathematics, in the field of calculus of several variables, the implicit function theorem says that for a suitable set of equations, some of the variables are defined as functions of the others. ...

Category: Calculus

Results from FactBites:

Inverse functions and differentiation - Wikipedia, the free encyclopedia (269 words)
	In mathematics, the inverse of a function y = f(x) is a function that, in some fashion, "undoes" the effect of f (see inverse function for a formal and detailed definition).
	Differentiation in calculus is the process of obtaining a derivative.
	It follows that functions with continuous derivative have inverses in a neighbourhood of every point where the derivative is non-zero.

Derivative - Wikipedia, the free encyclopedia (2201 words)
	Points on the graph of a function where the derivative is undefined or equals zero are called critical points or sometimes stationary points (in the case where the derivative equals zero).
	Perhaps the most natural situation is that of functions between differentiable manifolds; the derivative at a certain point then becomes a linear transformation between the corresponding tangent spaces and the derivative function becomes a map between the tangent bundles.
	For complex functions of a complex variable differentiability is a much stronger condition than that the real and imaginary part of the function are differentiable with respect to the real and imaginary part of the argument.

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