Inverse function theorem - Revision history

Vipul: /* Statement with symbols */

2013-01-20T18:39:25Z

Statement with symbols

← Older revision		Revision as of 18:39, 20 January 2013
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	! Version type !! Statement		! Version type !! Statement
	\|-		\|-
	\| specific point, named functions \|\| Suppose <math>f</math> is a [[function]] of one variable that is a [[one-one function]] and <math>a</math> is in the [[domain]] of <math>f</math>. Suppose <math>f</math> is continuous in an open interval containing <math>a</math>, [[differentiable function\|differentiable]] at <math>a</math> and <math>b = f(a)</math>. Suppose further that the [[fact about::derivative]] <math>f'(a)</math> is nonzero, i.e., <math>f'(a) \ne 0</math>. Then the [[fact about::inverse function]] <math>f^{-1}</math> is [[differentiable function\|differentiable]] at <math>b</math>, and further:<br><math>(f^{-1})'(b) = \frac{1}{f'(a)}</math>		\| specific point, named functions \|\| Suppose <math>f</math> is a [[function]] of one variable that is a [[one-one function]] and <math>a</math> is in the [[domain]] of <math>f</math>. Suppose <math>f</math> is continuous in an open interval containing <math>a</math> as well as [[differentiable function\|differentiable]] at <math>a</math>, and suppose <math>b = f(a)</math>. Suppose further that the [[fact about::derivative]] <math>f'(a)</math> is nonzero, i.e., <math>f'(a) \ne 0</math>. Then the [[fact about::inverse function]] <math>f^{-1}</math> is [[differentiable function\|differentiable]] at <math>b</math>, and further:<br><math>(f^{-1})'(b) = \frac{1}{f'(a)}</math>
	\|-		\|-
	\| generic point, named functions, point notation \|\| Suppose <math>f</math> is a [[function]] of one variable that is a [[one-one function]]. Then, the formula for the [[derivative]] of the [[inverse function]] is as follows: <br><math>\! (f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}</math><br> with the formula applicable at all points in the [[range]] of <math>f</math> for which <math>f</math> is continuous ''around'' the point, and <math>f'(f^{-1}(x))</math> exists and is nonzero.		\| generic point, named functions, point notation \|\| Suppose <math>f</math> is a [[function]] of one variable that is a [[one-one function]]. Then, the formula for the [[derivative]] of the [[inverse function]] is as follows: <br><math>\! (f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}</math><br> with the formula applicable at all points in the [[range]] of <math>f</math> for which <math>f</math> is continuous ''around'' the point and <math>f'(f^{-1}(x))</math> exists and is nonzero.
	\|-		\|-
	\| generic point, named functions, point-free notation \|\| Suppose <math>f</math> is a [[function]] of one variable that is a [[one-one function]]. Then, the formula for the [[derivative]] of the [[inverse function]] is as follows: <br><math>\! (f^{-1})'= \frac{1}{f' \circ f^{-1}}</math><br> with the formula applicable at all points in the [[range]] of <math>f</math> for which <math>f</math> is continuous around the point and <math>f'(f^{-1}(x))</math> exists and is nonzero.		\| generic point, named functions, point-free notation \|\| Suppose <math>f</math> is a [[function]] of one variable that is a [[one-one function]]. Then, the formula for the [[derivative]] of the [[inverse function]] is as follows: <br><math>\! (f^{-1})'= \frac{1}{f' \circ f^{-1}}</math><br> with the formula applicable at all points in the [[range]] of <math>f</math> for which <math>f</math> is continuous around the point and <math>f'(f^{-1}(x))</math> exists and is nonzero.

Vipul: /* Statement with symbols */

2013-01-20T18:28:29Z

Statement with symbols

← Older revision		Revision as of 18:28, 20 January 2013
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	! Version type !! Statement		! Version type !! Statement
	\|-		\|-
	\| specific point, named functions \|\| Suppose <math>f</math> is a [[function]] of one variable that is a [[one-one function]] and <math>a</math> is in the [[domain]] of <math>f</math>. Suppose <math>f</math> is [[differentiable function\|differentiable]] at <math>a</math> and <math>b = f(a)</math>. Suppose further that the [[fact about::derivative]] <math>f'(a)</math> is nonzero, i.e., <math>f'(a) \ne 0</math>. Then the [[fact about::inverse function]] <math>f^{-1}</math> is [[differentiable function\|differentiable]] at <math>b</math>, and further:<br><math>(f^{-1})'(b) = \frac{1}{f'(a)}</math>		\| specific point, named functions \|\| Suppose <math>f</math> is a [[function]] of one variable that is a [[one-one function]] and <math>a</math> is in the [[domain]] of <math>f</math>. Suppose <math>f</math> is continuous in an open interval containing <math>a</math>, [[differentiable function\|differentiable]] at <math>a</math> and <math>b = f(a)</math>. Suppose further that the [[fact about::derivative]] <math>f'(a)</math> is nonzero, i.e., <math>f'(a) \ne 0</math>. Then the [[fact about::inverse function]] <math>f^{-1}</math> is [[differentiable function\|differentiable]] at <math>b</math>, and further:<br><math>(f^{-1})'(b) = \frac{1}{f'(a)}</math>
	\|-		\|-
	\| generic point, named functions, point notation \|\| Suppose <math>f</math> is a [[function]] of one variable that is a [[one-one function]]. Then, the formula for the [[derivative]] of the [[inverse function]] is as follows: <br><math>\! (f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}</math><br> with the formula applicable at all points in the [[range]] of <math>f</math> for which <math>f'(f^{-1}(x))</math> exists and is nonzero.		\| generic point, named functions, point notation \|\| Suppose <math>f</math> is a [[function]] of one variable that is a [[one-one function]]. Then, the formula for the [[derivative]] of the [[inverse function]] is as follows: <br><math>\! (f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}</math><br> with the formula applicable at all points in the [[range]] of <math>f</math> for which <math>f</math> is continuous ''around'' the point, and <math>f'(f^{-1}(x))</math> exists and is nonzero.
	\|-		\|-
	\| generic point, named functions, point-free notation \|\| Suppose <math>f</math> is a [[function]] of one variable that is a [[one-one function]]. Then, the formula for the [[derivative]] of the [[inverse function]] is as follows: <br><math>\! (f^{-1})'= \frac{1}{f' \circ f^{-1}}</math><br> with the formula applicable at all points in the [[range]] of <math>f</math> for which <math>f'(f^{-1}(x))</math> exists and is nonzero.		\| generic point, named functions, point-free notation \|\| Suppose <math>f</math> is a [[function]] of one variable that is a [[one-one function]]. Then, the formula for the [[derivative]] of the [[inverse function]] is as follows: <br><math>\! (f^{-1})'= \frac{1}{f' \circ f^{-1}}</math><br> with the formula applicable at all points in the [[range]] of <math>f</math> for which <math>f</math> is continuous around the point and <math>f'(f^{-1}(x))</math> exists and is nonzero.
	\|-		\|-
	\| Pure Leibniz notation using dependent and independent variables \|\| Suppose <math>y</math> is a variable functionally dependent on <math>x</math>. Then, <math>\frac{dy}{dx} = \frac{1}{\frac{dx}{dy}}</math>		\| Pure Leibniz notation using dependent and independent variables \|\| Suppose <math>y</math> is a variable functionally dependent on <math>x</math>. Then, <math>\frac{dy}{dx} = \frac{1}{\frac{dx}{dy}}</math> (with domain caveats as above).
	\|}		\|}

Vipul: /* Examples */

2011-12-16T18:06:28Z

Examples

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 | undefined, but approaching <math>-\infty</math>, i.e., [[vertical tangent]] || zero || -- || Both decreasing. <math>f</math> is decreasing through <math>a</math> (with the rate of decrease peaking to <math>\infty</math>) and <math>f^{-1}</math> is decreasing through <math>b</math> (though the rate of decrease dips to zero because it's a [[point of inflection]] with horizontal tangent) || <math>f(x) := -x^{1/3}</math>, <math>f^{-1}(x) := -x^3</math>, <math>a = b = 0</math>
 |}
 ==Examples==

Vipul: /* Generic point examples */

2011-12-16T17:53:10Z

Generic point examples

← Older revision		Revision as of 17:53, 16 December 2011
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	\| [[tangent function]] <math>\tan</math> \|\| <math>(-\pi/2,\pi/2)</math> \|\| [[arc tangent function]] <math>\arctan</math> \|\| all real numbers \|\| [[secant-squared function]] <math>\sec^2</math> \|\| <math>\frac{1}{1 + x^2}</math> \|\| By the inverse function theorem, the derivative at <math>x</math> is <math>\frac{1}{\tan'(\arctan x)} = \frac{1}{\sec^2(\arctan x)}</math>. Use that <math>\sec^2 \theta = 1 + \tan^2\theta</math> and get <math>\sec^2(\arctan x) = 1 + x^2</math>.		\| [[tangent function]] <math>\tan</math> \|\| <math>(-\pi/2,\pi/2)</math> \|\| [[arc tangent function]] <math>\arctan</math> \|\| all real numbers \|\| [[secant-squared function]] <math>\sec^2</math> \|\| <math>\frac{1}{1 + x^2}</math> \|\| By the inverse function theorem, the derivative at <math>x</math> is <math>\frac{1}{\tan'(\arctan x)} = \frac{1}{\sec^2(\arctan x)}</math>. Use that <math>\sec^2 \theta = 1 + \tan^2\theta</math> and get <math>\sec^2(\arctan x) = 1 + x^2</math>.
	\|-		\|-
	\| [[natural logarithm]] <math>\ln</math> \|\| <math>(0,\infty)</math> \|\| [[exponential function]] <math>\exp</math> \|\| [[reciprocal function]] <math>x \mapsto 1/x</math> \|\| [[exponential function]] <math>\exp</math> \|\| By the inverse function theorem, the derivative at <math>x</math> is <math>\frac{1}{\ln'(\exp(x))} = \frac{1}{1/(\exp x)} = \exp(x)</math>		\| [[natural logarithm]] <math>\ln</math> \|\| <math>(0,\infty)</math> \|\| [[exponential function]] <math>\exp</math> \|\| all real numbers \|\| [[reciprocal function]] <math>x \mapsto 1/x</math> \|\| [[exponential function]] <math>\exp</math> \|\| By the inverse function theorem, the derivative at <math>x</math> is <math>\frac{1}{\ln'(\exp(x))} = \frac{1}{1/(\exp x)} = \exp(x)</math>
	\|}		\|}

Vipul: /* One-sided version */

2011-12-16T17:44:49Z

One-sided version

← Older revision		Revision as of 17:44, 16 December 2011
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Case for function !! Short version !! Long version (using specific point, named functions)		! Case for behavior of original function <math>f</math> at <math>a</math> !! Short version !! Long version (using specific point, named functions)
	\|-		\|-
	\| [[increasing function]] \|\| left hand derivative of <math>f^{-1}</math> is related to left hand derivative of <math>f</math> \|\| Suppose <math>f</math> is an [[increasing function]] ~~of one variable~~. Suppose <math>b = f(a)</math>. Suppose further that the [[fact about::derivative\|left hand derivative]] <math>f'_-(a)</math> is nonzero, i.e., <math>f'_-(a) \ne 0</math>. Then the [[fact about::inverse function]] <math>f^{-1}</math> is left differentiable at <math>b</math>, and further:<br><math>(f^{-1})'_-(b) = \frac{1}{f'_-(a)}</math>		\| [[increasing function]] from left \|\| left hand derivative of <math>f^{-1}</math> is related to left hand derivative of <math>f</math> \|\| Suppose <math>f</math> is an [[increasing function]] from the left at a point <math>a</math>. Suppose <math>b = f(a)</math>. Suppose further that the [[fact about::derivative\|left hand derivative]] <math>f'_-(a)</math> is nonzero, i.e., <math>f'_-(a) \ne 0</math>. Then the [[fact about::inverse function]] <math>f^{-1}</math> is left differentiable at <math>b</math>, and further:<br><math>(f^{-1})'_-(b) = \frac{1}{f'_-(a)}</math>
	\|-		\|-
	\| [[increasing function]] \|\| right hand derivative of <math>f^{-1}</math> is related to right hand derivative of <math>f</math> \|\| Suppose <math>f</math> is an [[increasing function]] of ~~one variable~~. Suppose <math>b = f(a)</math>. Suppose further that the [[fact about::derivative\|right hand derivative]] <math>f'_+(a)</math> is nonzero, i.e., <math>f'_+(a) \ne 0</math>. Then the [[fact about::inverse function]] <math>f^{-1}</math> is right differentiable at <math>b</math>, and further:<br><math>(f^{-1})'_+(b) = \frac{1}{f'_+(a)}</math>		\| [[increasing function]] from right \|\| right hand derivative of <math>f^{-1}</math> is related to right hand derivative of <math>f</math> \|\| Suppose <math>f</math> is an [[increasing function]] from the right at a point <math>a</math> (in other words, <math>f</math> increases on the immediate right of <math>a</math>). Suppose <math>b = f(a)</math>. Suppose further that the [[fact about::derivative\|right hand derivative]] <math>f'_+(a)</math> is nonzero, i.e., <math>f'_+(a) \ne 0</math>. Then the [[fact about::inverse function]] <math>f^{-1}</math> is right differentiable at <math>b</math>, and further:<br><math>(f^{-1})'_+(b) = \frac{1}{f'_+(a)}</math>
	\|-		\|-
	\| [[decreasing function]] \|\| right hand derivative of <math>f^{-1}</math> is related to left hand derivative of <math>f</math> \|\| Suppose <math>f</math> is a [[decreasing function]] ~~of one variable~~. Suppose <math>b = f(a)</math>. Suppose further that the [[fact about::derivative\|left hand derivative]] <math>f'_-(a)</math> is nonzero, i.e., <math>f'_-(a) \ne 0</math>. Then the [[fact about::inverse function]] <math>f^{-1}</math> is right differentiable at <math>b</math>, and further:<br><math>(f^{-1})'_+(b) = \frac{1}{f'_-(a)}</math>		\| [[decreasing function]] from left\|\| right hand derivative of <math>f^{-1}</math> is related to left hand derivative of <math>f</math> \|\| Suppose <math>f</math> is a [[decreasing function]] on the left at a point <math>a</math>. Suppose <math>b = f(a)</math>. Suppose further that the [[fact about::derivative\|left hand derivative]] <math>f'_-(a)</math> is nonzero, i.e., <math>f'_-(a) \ne 0</math>. Then the [[fact about::inverse function]] <math>f^{-1}</math> is right differentiable at <math>b</math>, and further:<br><math>(f^{-1})'_+(b) = \frac{1}{f'_-(a)}</math>
	\|-		\|-
	\| [[decreasing function]] \|\| left hand derivative of <math>f^{-1}</math> is related to right hand derivative of <math>f</math> \|\| Suppose <math>f</math> is a [[decreasing function]] ~~of one variable~~. Suppose <math>b = f(a)</math>. Suppose further that the [[fact about::derivative\|right hand derivative]] <math>f'_+(a)</math> is nonzero, i.e., <math>f'_+(a) \ne 0</math>. Then the [[fact about::inverse function]] <math>f^{-1}</math> is left differentiable at <math>b</math>, and further:<br><math>(f^{-1})'_-(b) = \frac{1}{f'_+(a)}</math>		\| [[decreasing function]] from right\|\| left hand derivative of <math>f^{-1}</math> is related to right hand derivative of <math>f</math> \|\| Suppose <math>f</math> is a [[decreasing function]] from the right at a point <math>a</math>. Suppose <math>b = f(a)</math>. Suppose further that the [[fact about::derivative\|right hand derivative]] <math>f'_+(a)</math> is nonzero, i.e., <math>f'_+(a) \ne 0</math>. Then the [[fact about::inverse function]] <math>f^{-1}</math> is left differentiable at <math>b</math>, and further:<br><math>(f^{-1})'_-(b) = \frac{1}{f'_+(a)}</math>
	\|}		\|}

			Some additional notes:

			* For a point in the interior of the domain at which the function is continuous, being increasing on the immediate left forces the function to be increasing on the immediate right, and vice versa. Similarly for decreasing.
			* More generally, a continuous one-one function on an interval must be either increasing through the interval or decreasing throughout the interval.

			We have been more specific in our statements in the table above to allow for the possibility of piecewise defined functions with discontinuities as well as to tackle the issue of interval endpoints where only one-sided notions make sense.

	===Infinity-sensitive versions===		===Infinity-sensitive versions===

Vipul at 17:41, 16 December 2011

2011-12-16T17:41:21Z

@@ Line 57: / Line 57: @@
 |-
 | undefined, but approaching <math>-\infty</math>, i.e., [[vertical tangent]] || zero || -- || Both decreasing. <math>f</math> is decreasing through <math>a</math> (with the rate of decrease peaking to <math>\infty</math>) and <math>f^{-1}</math> is decreasing through <math>b</math> (though the rate of decrease dips to zero because it's a [[point of inflection]] with horizontal tangent) || <math>f(x) := -x^{1/3}</math>, <math>f^{-1}(x) := -x^3</math>, <math>a = b = 0</math>
 |}

Vipul: /* One-sided version */

2011-12-16T17:34:19Z

One-sided version

← Older revision		Revision as of 17:34, 16 December 2011
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	! Case for function !! Short version !! Long version (using specific point, named functions)		! Case for function !! Short version !! Long version (using specific point, named functions)
	\|-		\|-
	\| [[increasing function]] \|\| left hand derivative of <math>f^{-1}</math> is related to left hand derivative of <math>f</math> \|\| Suppose <math>f</math> is a [[function]] of one variable ~~that is a [[one-one function]] and <math>a</math> is in the [[domain]] of <math>f</math>~~. Suppose <math>b = f(a)</math>. Suppose further that the [[fact about::derivative\|left hand derivative]] <math>f'_-(a)</math> is nonzero, i.e., <math>f'_-(a) \ne 0</math>. Then the [[fact about::inverse function]] <math>f^{-1}</math> is left differentiable at <math>b</math>, and further:<br><math>(f^{-1})'_-(b) = \frac{1}{f'_-(a)}</math>		\| [[increasing function]] \|\| left hand derivative of <math>f^{-1}</math> is related to left hand derivative of <math>f</math> \|\| Suppose <math>f</math> is an [[increasing function]] of one variable. Suppose <math>b = f(a)</math>. Suppose further that the [[fact about::derivative\|left hand derivative]] <math>f'_-(a)</math> is nonzero, i.e., <math>f'_-(a) \ne 0</math>. Then the [[fact about::inverse function]] <math>f^{-1}</math> is left differentiable at <math>b</math>, and further:<br><math>(f^{-1})'_-(b) = \frac{1}{f'_-(a)}</math>
	\|-		\|-
	\| [[increasing function]] \|\| right hand derivative of <math>f^{-1}</math> is related to right hand derivative of <math>f</math> \|\| Suppose <math>f</math> is a [[function]] of one variable ~~that is a [[one-one function]] and <math>a</math> is in the [[domain]] of <math>f</math>~~. Suppose <math>b = f(a)</math>. Suppose further that the [[fact about::derivative\|right hand derivative]] <math>f'_+(a)</math> is nonzero, i.e., <math>f'_+(a) \ne 0</math>. Then the [[fact about::inverse function]] <math>f^{-1}</math> is right differentiable at <math>b</math>, and further:<br><math>(f^{-1})'_+(b) = \frac{1}{f'_+(a)}</math>		\| [[increasing function]] \|\| right hand derivative of <math>f^{-1}</math> is related to right hand derivative of <math>f</math> \|\| Suppose <math>f</math> is an [[increasing function]] of one variable. Suppose <math>b = f(a)</math>. Suppose further that the [[fact about::derivative\|right hand derivative]] <math>f'_+(a)</math> is nonzero, i.e., <math>f'_+(a) \ne 0</math>. Then the [[fact about::inverse function]] <math>f^{-1}</math> is right differentiable at <math>b</math>, and further:<br><math>(f^{-1})'_+(b) = \frac{1}{f'_+(a)}</math>
	\|-		\|-
	\| [[decreasing function]] \|\| right hand derivative of <math>f^{-1}</math> is related to left hand derivative of <math>f</math> \|\|		\| [[decreasing function]] \|\| right hand derivative of <math>f^{-1}</math> is related to left hand derivative of <math>f</math> \|\| Suppose <math>f</math> is a [[decreasing function]] of one variable. Suppose <math>b = f(a)</math>. Suppose further that the [[fact about::derivative\|left hand derivative]] <math>f'_-(a)</math> is nonzero, i.e., <math>f'_-(a) \ne 0</math>. Then the [[fact about::inverse function]] <math>f^{-1}</math> is right differentiable at <math>b</math>, and further:<br><math>(f^{-1})'_+(b) = \frac{1}{f'_-(a)}</math>
	\|-		\|-
	\| [[decreasing function]] \|\| left hand derivative of <math>f^{-1}</math> is related to right hand derivative of <math>f</math> \|\|		\| [[decreasing function]] \|\| left hand derivative of <math>f^{-1}</math> is related to right hand derivative of <math>f</math> \|\| Suppose <math>f</math> is a [[decreasing function]] of one variable. Suppose <math>b = f(a)</math>. Suppose further that the [[fact about::derivative\|right hand derivative]] <math>f'_+(a)</math> is nonzero, i.e., <math>f'_+(a) \ne 0</math>. Then the [[fact about::inverse function]] <math>f^{-1}</math> is left differentiable at <math>b</math>, and further:<br><math>(f^{-1})'_-(b) = \frac{1}{f'_+(a)}</math>
	\|}		\|}

Vipul: /* Statement */

2011-12-16T17:31:02Z

Statement

← Older revision		Revision as of 17:31, 16 December 2011
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	==Statement==		==Statement==

	===~~Simple version at a specific point~~===		===Verbal statement===

	~~Suppose <math>f</math> is a [[function]]~~ of ~~one variable that is a [[one-one~~ function~~]] and <math>~~a~~</math> is in~~ the ~~[[domain]]~~ of ~~<math>f</math>. Suppose <math>f</math> is [differentiable~~ function~~\|differentiable]]~~ at ~~<math>a</math> and <math>b = f(a)</math>. Suppose further that the [[fact about::derivative]] <math>f'(a)</math> is nonzero, i.e., <math>f'(a) \ne 0</math>~~. ~~Then:~~		The derivative of the inverse function at a point equals the reciprocal of the derivative of the function at its inverse image point.

	~~The [[fact about::inverse function]] <math>f^{-1}</math> is [[differentiable function\|differentiable]] at <math>b</math>, and further:~~		===Statement with symbols===

	<math>(f^{-1})'(b) = \frac{1}{f'(a)}</math>		{\| class="sortable" border="1"
			! Version type !! Statement
			\|-
			\| specific point, named functions \|\| Suppose <math>f</math> is a [[function]] of one variable that is a [[one-one function]] and <math>a</math> is in the [[domain]] of <math>f</math>. Suppose <math>f</math> is [[differentiable function\|differentiable]] at <math>a</math> and <math>b = f(a)</math>. Suppose further that the [[fact about::derivative]] <math>f'(a)</math> is nonzero, i.e., <math>f'(a) \ne 0</math>. Then the [[fact about::inverse function]] <math>f^{-1}</math> is [[differentiable function\|differentiable]] at <math>b</math>, and further:<br><math>(f^{-1})'(b) = \frac{1}{f'(a)}</math>
			\|-
			\| generic point, named functions, point notation \|\| Suppose <math>f</math> is a [[function]] of one variable that is a [[one-one function]]. Then, the formula for the [[derivative]] of the [[inverse function]] is as follows: <br><math>\! (f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}</math><br> with the formula applicable at all points in the [[range]] of <math>f</math> for which <math>f'(f^{-1}(x))</math> exists and is nonzero.
			\|-
			\| generic point, named functions, point-free notation \|\| Suppose <math>f</math> is a [[function]] of one variable that is a [[one-one function]]. Then, the formula for the [[derivative]] of the [[inverse function]] is as follows: <br><math>\! (f^{-1})'= \frac{1}{f' \circ f^{-1}}</math><br> with the formula applicable at all points in the [[range]] of <math>f</math> for which <math>f'(f^{-1}(x))</math> exists and is nonzero.
			\|-
			\| Pure Leibniz notation using dependent and independent variables \|\| Suppose <math>y</math> is a variable functionally dependent on <math>x</math>. Then, <math>\frac{dy}{dx} = \frac{1}{\frac{dx}{dy}}</math>
			\|}

	~~===Simple version at a~~ generic point~~===~~		{{generic point specific point confusion}}

	~~Suppose <math>f</math> is a [[function]] of one variable that is a [[one~~-~~one function]]. Then, the formula for the [[derivative]] of the [[inverse function]] is as follows:~~		===One-sided version===

	~~<math>(f^{~~-~~1})'(x) = \frac{1}{f'(f^{-1}(x))}</math>~~		One-sided versions exist, but we need to be careful about issues of left and right. We state the two cases:

	~~with the formula applicable at all points~~ in the [[~~range~~]] of <math>f</math> ~~for which~~ <math>f'(f^{-1}(x))</math> ~~exists~~ and is nonzero.		{\| class="sortable" border="1"
			! Case for function !! Short version !! Long version (using specific point, named functions)
			\|-
			\| [[increasing function]] \|\| left hand derivative of <math>f^{-1}</math> is related to left hand derivative of <math>f</math> \|\| Suppose <math>f</math> is a [[function]] of one variable that is a [[one-one function]] and <math>a</math> is in the [[domain]] of <math>f</math>. Suppose <math>b = f(a)</math>. Suppose further that the [[fact about::derivative\|left hand derivative]] <math>f'_-(a)</math> is nonzero, i.e., <math>f'_-(a) \ne 0</math>. Then the [[fact about::inverse function]] <math>f^{-1}</math> is left differentiable at <math>b</math>, and further:<br><math>(f^{-1})'_-(b) = \frac{1}{f'_-(a)}</math>
			\|-
			\| [[increasing function]] \|\| right hand derivative of <math>f^{-1}</math> is related to right hand derivative of <math>f</math> \|\| Suppose <math>f</math> is a [[function]] of one variable that is a [[one-one function]] and <math>a</math> is in the [[domain]] of <math>f</math>. Suppose <math>b = f(a)</math>. Suppose further that the [[fact about::derivative\|right hand derivative]] <math>f'_+(a)</math> is nonzero, i.e., <math>f'_+(a) \ne 0</math>. Then the [[fact about::inverse function]] <math>f^{-1}</math> is right differentiable at <math>b</math>, and further:<br><math>(f^{-1})'_+(b) = \frac{1}{f'_+(a)}</math>
			\|-
			\| [[decreasing function]] \|\| right hand derivative of <math>f^{-1}</math> is related to left hand derivative of <math>f</math> \|\|
			\|-
			\| [[decreasing function]] \|\| left hand derivative of <math>f^{-1}</math> is related to right hand derivative of <math>f</math> \|\|
			\|}

	===~~One~~-~~sided~~ versions===		===Infinity-sensitive versions===

	~~{{fillin}}~~		The following version accounts for the infinity cases. We provide only the ''specific point, named functions'' version. Assume that <math>f</math> is a one-one function that is continuous at a point <math>a</math> in its domain, with <math>b = f(a)</math>. There are six cases of interest:

	~~===Infinity~~-~~sensitive versions==~~=

	{{~~fillin~~}}		{\| class="sortable" border="1"
			! Case for <math>f'(a)</math> !! Case for <math>(f^{-1})'(b)</math> !! Relation between them !! Increase/decrease? !! Example
			\|-
			\| undefined, but approaching <math>+\infty</math> \|\| zero \|\| -- \|\| Both increasing. <math>f</math> is increasing through <math>a</math> (with the rate of increase peaking to <math>\infty</math>) and <math>f^{-1}</math> is increasing through <math>b</math> (though the rate of increase dips to zero because it's a [[point of inflection]] with horizontal tangent). \|\| <math>f(x) := x^{1/3}</math>, <math>f^{-1}(x) := x^3</math>, <math>a = b = 0</math>
			\|-
			\| positive \|\| positive \|\| reciprocals of each other. \|\| Both increasing. <math>f</math> is increasing through <math>a</math> and <math>f^{-1}</math> is increasing through <math>b</math>. \|\| <math>f(x) := x^3</math>, <math>f^{-1}(x) := x^{1/3}</math>, <math>a = b = 1</math>
			\|-
			\| zero \|\| undefined, but approaching <math>+\infty</math>, i.e., [[vertical tangent]] \|\| -- \|\| Both increasing. <math>f</math> is increasing through <math>a</math> (though the rate of increase dips to zero because it's a [[point of inflection]] with horizontal tangent) and <math>f^{-1}</math> is increasing through <math>b</math> (with the rate of increase peaking to <math>\infty</math>). \|\| <math>f(x) := x^3</math>, <math>f^{-1}(x) := x^{1/3}</math>, <math>a = b = 0</math>
			\|-
			\| zero \|\| undefined, but approaching <math>-\infty</math>, i.e., [[vertical tangent]] \|\| -- \|\| Both decreasing. <math>f</math> is decreasing through <math>a</math> (though the rate of decrease dips to zero because it's a [[point of inflection]] with horizontal tangent) and <math>f^{-1}</math> is decreasing through <math>b</math> (with the rate of decrease peaking to <math>\infty</math>). \|\| <math>f(x) := -x^3</math>, <math>f^{-1}(x) := -x^{1/3}</math>, <math>a = b = 0</math>
			\|-
			\| negative \|\| negative \|\| reciprocals of each other \|\| Both decreasing. <math>f</math> is decreasing through <math>a</math> and <math>f^{-1}</math> is decreasing through <math>b</math>. \|\| <math>f(x) := -x^3</math>, <math>f^{-1}(x) := -x^{1/3}</math>, <math>a = 1,b = -1</math>
			\|-
			\| undefined, but approaching <math>-\infty</math>, i.e., [[vertical tangent]] \|\| zero \|\| -- \|\| Both decreasing. <math>f</math> is decreasing through <math>a</math> (with the rate of decrease peaking to <math>\infty</math>) and <math>f^{-1}</math> is decreasing through <math>b</math> (though the rate of decrease dips to zero because it's a [[point of inflection]] with horizontal tangent) \|\| <math>f(x) := -x^{1/3}</math>, <math>f^{-1}(x) := -x^3</math>, <math>a = b = 0</math>
			\|}

Vipul at 22:39, 21 September 2011

2011-09-21T22:39:43Z

← Older revision		Revision as of 22:39, 21 September 2011
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			{{differentiation rule}}

	==Statement==		==Statement==

Vipul: Created page with "==Statement== ===Simple version at a specific point=== Suppose $f$ is a function of one variable that is a one-one function and $a$ is in the [[do..."

2011-09-21T22:38:12Z

Created page with "==Statement== ===Simple version at a specific point=== Suppose <math>f</math> is a function of one variable that is a one-one function and <math>a</math> is in the [[do..."

New page

==Statement==

===Simple version at a specific point===

Suppose <math>f</math> is a [[function]] of one variable that is a [[one-one function]] and <math>a</math> is in the [[domain]] of <math>f</math>. Suppose <math>f</math> is [differentiable function|differentiable]] at <math>a</math> and <math>b = f(a)</math>. Suppose further that the [[fact about::derivative]] <math>f'(a)</math> is nonzero, i.e., <math>f'(a) \ne 0</math>. Then:

The [[fact about::inverse function]] <math>f^{-1}</math> is [[differentiable function|differentiable]] at <math>b</math>, and further:

<math>(f^{-1})'(b) = \frac{1}{f'(a)}</math>

===Simple version at a generic point===

Suppose <math>f</math> is a [[function]] of one variable that is a [[one-one function]]. Then, the formula for the [[derivative]] of the [[inverse function]] is as follows:

<math>(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}</math>

with the formula applicable at all points in the [[range]] of <math>f</math> for which <math>f'(f^{-1}(x))</math> exists and is nonzero.

===One-sided versions===

{{fillin}}

===Infinity-sensitive versions===

{{fillin}}

Inverse function theorem - Revision history

Vipul: /* Statement with symbols */

Vipul: /* Statement with symbols */

Vipul: /* Examples */

Vipul: /* Generic point examples */

Vipul: /* One-sided version */

Vipul at 17:41, 16 December 2011

Vipul: /* One-sided version */

Vipul: /* Statement */

Vipul at 22:39, 21 September 2011

Vipul: Created page with "==Statement== ===Simple version at a specific point=== Suppose f is a function of one variable that is a one-one function and a is in the [[do..."

Vipul: Created page with "==Statement== ===Simple version at a specific point=== Suppose $f$ is a function of one variable that is a one-one function and $a$ is in the [[do..."