Difference between revisions of "Derivatives of Inverse Functions"

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(Added links for PowerPoint and worksheet)
 
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In mathematics, the '''inverse''' of a function <math>y = f(x)</math> is a function that, in some fashion, "undoes" the effect of <math>f</math>. The inverse of <math>f</math> is denoted as <math>f^{-1}</math>, where <math>f^{-1}(y) = x</math> if and only if <math>f(x) = y</math>.
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Their two derivatives, assuming they exist, are reciprocal, as the Leibniz notation suggests; that is:
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:<math>\frac{dx}{dy}\,\cdot\, \frac{dy}{dx} = 1.</math>
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This relation is obtained by differentiating the equation <math>f^{-1}(y)=x</math> in terms of <math>x</math> and applying the chain rule, yielding that:
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:<math>\frac{dx}{dy}\,\cdot\, \frac{dy}{dx} = \frac{dx}{dx}</math>
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considering that the derivative of <math>x</math> with respect to ''<math>x</math>'' is 1.
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Writing explicitly the dependence of <math>y</math> on <math>x</math>, and the point at which the differentiation takes place, the formula for the derivative of the inverse becomes (in Lagrange's notation):
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:<math>\left[f^{-1}\right]'(a)=\frac{1}{f'\left( f^{-1}(a) \right)}</math>.
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This formula holds in general whenever <math>f</math> is continuous and injective on an interval <math>I</math>, with <math>f</math> being differentiable at <math>f^{-1}(a)</math>(<math>\in I</math>) and where <math>f'(f^{-1}(a)) \ne 0</math>. The same formula is also equivalent to the expression
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:<math>\mathcal{D}\left[f^{-1}\right]=\frac{1}{(\mathcal{D} f)\circ \left(f^{-1}\right)},</math>
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where <math>\mathcal{D}</math> denotes the unary derivative operator (on the space of functions) and <math>\circ</math> denotes function composition.
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Geometrically, a function and inverse function have graphs that are reflections, in the line <math>y=x</math>. This reflection operation turns the gradient of any line into its reciprocal.
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Assuming that <math>f</math> has an inverse in a neighbourhood of <math>x</math> and that its derivative at that point is non-zero, its inverse is guaranteed to be differentiable at <math>x</math> and have a derivative given by the above formula.
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==Resources==
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* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Derivatives%20of%20Inverse%20Functions_/MAT1214-3.7DerivativesOfInverseFunctionsPwPt.pptx  Derivatives of Inverse Functions] PowerPoint file created by Dr. Sara Shirinkam, UTSA.
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Derivatives%20of%20Inverse%20Functions_/MAT1214-3.7DerivativesOfInverseFunctionsPwPt.pptx  Derivatives of Inverse Functions] PowerPoint file created by Dr. Sara Shirinkam, UTSA.
  
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Derivatives%20of%20Inverse%20Functions_/MAT1214-3.7DerivativesOfInverseFunctionsWS1.pdf  Derivatives of Inverse Functions Worksheet]
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Derivatives%20of%20Inverse%20Functions_/MAT1214-3.7DerivativesOfInverseFunctionsWS1.pdf  Derivatives of Inverse Functions Worksheet]
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* [https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Calculus_(OpenStax)/03%3A_Derivatives/3.7%3A_Derivatives_of_Inverse_Functions Derivatives of Inverse Function], Mathematics LibreTexts

Revision as of 16:33, 29 September 2021

In mathematics, the inverse of a function is a function that, in some fashion, "undoes" the effect of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} . The inverse of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is denoted as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(y) = x} if and only if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = y} .

Their two derivatives, assuming they exist, are reciprocal, as the Leibniz notation suggests; that is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dx}{dy}\,\cdot\, \frac{dy}{dx} = 1.}

This relation is obtained by differentiating the equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(y)=x} in terms of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and applying the chain rule, yielding that:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dx}{dy}\,\cdot\, \frac{dy}{dx} = \frac{dx}{dx}}

considering that the derivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} with respect to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is 1.

Writing explicitly the dependence of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} , and the point at which the differentiation takes place, the formula for the derivative of the inverse becomes (in Lagrange's notation):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[f^{-1}\right]'(a)=\frac{1}{f'\left( f^{-1}(a) \right)}} .

This formula holds in general whenever Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is continuous and injective on an interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} , with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} being differentiable at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(a)} (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \in I} ) and where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(f^{-1}(a)) \ne 0} . The same formula is also equivalent to the expression

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{D}\left[f^{-1}\right]=\frac{1}{(\mathcal{D} f)\circ \left(f^{-1}\right)},}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{D}} denotes the unary derivative operator (on the space of functions) and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \circ} denotes function composition.

Geometrically, a function and inverse function have graphs that are reflections, in the line Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=x} . This reflection operation turns the gradient of any line into its reciprocal.

Assuming that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} has an inverse in a neighbourhood of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and that its derivative at that point is non-zero, its inverse is guaranteed to be differentiable at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and have a derivative given by the above formula.


Resources

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