Difference between revisions of "Derivatives of Inverse Functions"

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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.
  
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==Examples==
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* <math>y = x^2</math> (for positive {{Mvar|x}}) has inverse <math>x = \sqrt{y}</math>.
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:<math> \frac{dy}{dx} = 2x
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\mbox{ }\mbox{ }\mbox{ }\mbox{ };
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\mbox{ }\mbox{ }\mbox{ }\mbox{ }
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\frac{dx}{dy} = \frac{1}{2\sqrt{y}}=\frac{1}{2x} </math>
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:<math>\frac{dy}{dx}\,\cdot\,\frac{dx}{dy} = 2x \cdot\frac{1}{2x}  =  1.</math>
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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.
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* <math>y = e^x</math> (for real {{Mvar|x}}) has inverse <math>x = \ln{y}</math> (for positive <math>y</math>)
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 +
:<math> \frac{dy}{dx} = e^x
 +
\mbox{ }\mbox{ }\mbox{ }\mbox{ };
 +
\mbox{ }\mbox{ }\mbox{ }\mbox{ }
 +
\frac{dx}{dy} = \frac{1}{y} </math>
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:<math> \frac{dy}{dx}\,\cdot\,\frac{dx}{dy}  =  e^x \cdot \frac{1}{y}  =  \frac{e^x}{e^x}  =  1 </math>
  
  

Revision as of 09:31, 28 October 2021

Rule:


Example for arbitrary :

In mathematics, the inverse of a function is a function that, in some fashion, "undoes" the effect of . The inverse of is denoted as , where if and only if .

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

This relation is obtained by differentiating the equation in terms of and applying the chain rule, yielding that:

considering that the derivative of with respect to is 1.

Writing explicitly the dependence of on , and the point at which the differentiation takes place, the formula for the derivative of the inverse becomes (in Lagrange's notation):

.

This formula holds in general whenever is continuous and injective on an interval , with being differentiable at () and where . The same formula is also equivalent to the expression

where denotes the unary derivative operator (on the space of functions) and denotes function composition.

Geometrically, a function and inverse function have graphs that are reflections, in the line . This reflection operation turns the gradient of any line into its reciprocal.

Assuming that has an inverse in a neighbourhood of and that its derivative at that point is non-zero, its inverse is guaranteed to be differentiable at and have a derivative given by the above formula.

Examples

  • (for positive x) has inverse .

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

  • (for real x) has inverse (for positive )


Resources