Derivatives of Inverse Functions - Revision history

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2021-10-28T15:31:55Z

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Lila at 15:31, 28 October 2021

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Lila at 15:29, 28 October 2021

2021-10-28T15:29:41Z

Lila at 22:33, 29 September 2021

2021-09-29T22:33:27Z

James.kercheville: Added links for PowerPoint and worksheet

2020-08-13T23:29:47Z

Added links for PowerPoint and worksheet

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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_/MAT1214-3.7DerivativesOfInverseFunctionsWS1.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		* [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
		+
		+	==Licensing==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikipedia.org/wiki/Inverse_functions_and_differentiation Inverse functions and differentiation, Wikipedia] under a CC BY-SA license

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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.		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.

		+	==Examples==
		+
		+	* <math>y = x^2</math> (for positive {{Mvar\|x}}) has inverse <math>x = \sqrt{y}</math>.
		+
		+	:<math> \frac{dy}{dx} = 2x
		+	\mbox{ }\mbox{ }\mbox{ }\mbox{ };
		+	\mbox{ }\mbox{ }\mbox{ }\mbox{ }
		+	\frac{dx}{dy} = \frac{1}{2\sqrt{y}}=\frac{1}{2x} </math>
		+
		+	:<math>\frac{dy}{dx}\,\cdot\,\frac{dx}{dy} = 2x \cdot\frac{1}{2x} = 1.</math>
		+
		+	At <math>x=0</math>, however, there is a problem: the graph of the square root function becomes vertical, corresponding to a horizontal tangent for the square function.
		+
		+	* <math>y = e^x</math> (for real {{Mvar\|x}}) has inverse <math>x = \ln{y}</math> (for positive <math>y</math>)
		+
		+	:<math> \frac{dy}{dx} = e^x
		+	\mbox{ }\mbox{ }\mbox{ }\mbox{ };
		+	\mbox{ }\mbox{ }\mbox{ }\mbox{ }
		+	\frac{dx}{dy} = \frac{1}{y} </math>
		+
		+	:<math> \frac{dy}{dx}\,\cdot\,\frac{dx}{dy} = e^x \cdot \frac{1}{y} = \frac{e^x}{e^x} = 1 </math>

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		+	[[File:Umkehrregel 2.png\|thumb\|right\|250px\|Rule:<br><math>{\color{CornflowerBlue}{f'}}(x) = \frac{1}{{\color{Salmon}{(f^{-1})'}}({\color{Blue}{f}}(x))}</math><br><br>Example for arbitrary <math>x_0 \approx 5.8</math>:<br><math>{\color{CornflowerBlue}{f'}}(x_0) = \frac{1}{4}</math><br><math>{\color{Salmon}{(f^{-1})'}}({\color{Blue}{f}}(x_0)) = 4~</math>]]
		+
	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>.		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>.

← Older revision		Revision as of 22:33, 29 September 2021
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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>.
		+
		+	Their two derivatives, assuming they exist, are reciprocal, as the Leibniz notation suggests; that is:
		+
		+	:<math>\frac{dx}{dy}\,\cdot\, \frac{dy}{dx} = 1.</math>
		+
		+	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:
		+
		+	:<math>\frac{dx}{dy}\,\cdot\, \frac{dy}{dx} = \frac{dx}{dx}</math>
		+
		+	considering that the derivative of <math>x</math> with respect to ''<math>x</math>'' is 1.
		+
		+	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):
		+
		+	:<math>\left[f^{-1}\right]'(a)=\frac{1}{f'\left( f^{-1}(a) \right)}</math>.
		+
		+	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
		+
		+	:<math>\mathcal{D}\left[f^{-1}\right]=\frac{1}{(\mathcal{D} f)\circ \left(f^{-1}\right)},</math>
		+
		+	where <math>\mathcal{D}</math> denotes the unary derivative operator (on the space of functions) and <math>\circ</math> denotes function composition.
		+
		+	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.
		+
		+	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.
		+
		+
		+
		+	==Resources==
		+
	* [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]
		+
		+	* [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