Difference between revisions of "Inverse function theorem"

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(Statement)
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==Statement==
 
==Statement==
  
===Simple version at a specific point===
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===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:
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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:
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===Statement with symbols===
  
<math>(f^{-1})'(b) = \frac{1}{f'(a)}</math>
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{| class="sortable" border="1"
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! Version type !! Statement
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|-
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| 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>
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|-
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| 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.
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|-
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| 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.
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|-
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| 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>
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|}
  
===Simple version at a generic point===
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{{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:
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===One-sided version===
  
<math>(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}</math>
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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.
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{| class="sortable" border="1"
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! Case for function !! Short version !! Long version (using specific point, named functions)
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|-
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| [[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>
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|-
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| [[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>
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|-
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| [[decreasing function]] || right hand derivative of <math>f^{-1}</math> is related to left hand derivative of <math>f</math> ||
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|-
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| [[decreasing function]] || left hand derivative of <math>f^{-1}</math> is related to right hand derivative of <math>f</math> ||
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|}
  
===One-sided versions===
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===Infinity-sensitive versions===
  
{{fillin}}
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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}}
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{| class="sortable" border="1"
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! Case for <math>f'(a)</math> !! Case for <math>(f^{-1})'(b)</math> !! Relation between them !! Increase/decrease? !! Example
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|-
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| 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>
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|-
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| 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>
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|-
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| 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>
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|-
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| 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>
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|-
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| 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>
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|-
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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>
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|}

Revision as of 17:31, 16 December 2011

This article is about a differentiation rule, i.e., a rule for differentiating a function expressed in terms of other functions whose derivatives are known.
View other differentiation rules

Statement

Verbal statement

The derivative of the inverse function at a point equals the reciprocal of the derivative of the function at its inverse image point.

Statement with symbols

Version type Statement
specific point, named functions Suppose f is a function of one variable that is a one-one function and a is in the domain of f. Suppose f is differentiable at a and b = f(a). Suppose further that the derivative f'(a) is nonzero, i.e., f'(a) \ne 0. Then the inverse function f^{-1} is differentiable at b, and further:
(f^{-1})'(b) = \frac{1}{f'(a)}
generic point, named functions, point notation Suppose f 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:
\! (f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}
with the formula applicable at all points in the range of f for which f'(f^{-1}(x)) exists and is nonzero.
generic point, named functions, point-free notation Suppose f 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:
\! (f^{-1})'= \frac{1}{f' \circ f^{-1}}
with the formula applicable at all points in the range of f for which f'(f^{-1}(x)) exists and is nonzero.
Pure Leibniz notation using dependent and independent variables Suppose y is a variable functionally dependent on x. Then, \frac{dy}{dx} = \frac{1}{\frac{dx}{dy}}
MORE ON THE WAY THIS DEFINITION OR FACT IS PRESENTED: We first present the version that deals with a specific point (typically with a \{ \}_0 subscript) in the domain of the relevant functions, and then discuss the version that deals with a point that is free to move in the domain, by dropping the subscript. Why do we do this?
The purpose of the specific point version is to emphasize that the point is fixed for the duration of the definition, i.e., it does not move around while we are defining the construct or applying the fact. However, the definition or fact applies not just for a single point but for all points satisfying certain criteria, and thus we can get further interesting perspectives on it by varying the point we are considering. This is the purpose of the second, generic point version.

One-sided version

One-sided versions exist, but we need to be careful about issues of left and right. We state the two cases:

Case for function Short version Long version (using specific point, named functions)
increasing function left hand derivative of f^{-1} is related to left hand derivative of f Suppose f is a function of one variable that is a one-one function and a is in the domain of f. Suppose b = f(a). Suppose further that the left hand derivative f'_-(a) is nonzero, i.e., f'_-(a) \ne 0. Then the inverse function f^{-1} is left differentiable at b, and further:
(f^{-1})'_-(b) = \frac{1}{f'_-(a)}
increasing function right hand derivative of f^{-1} is related to right hand derivative of f Suppose f is a function of one variable that is a one-one function and a is in the domain of f. Suppose b = f(a). Suppose further that the right hand derivative f'_+(a) is nonzero, i.e., f'_+(a) \ne 0. Then the inverse function f^{-1} is right differentiable at b, and further:
(f^{-1})'_+(b) = \frac{1}{f'_+(a)}
decreasing function right hand derivative of f^{-1} is related to left hand derivative of f
decreasing function left hand derivative of f^{-1} is related to right hand derivative of f

Infinity-sensitive versions

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

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