Editing Inverse function rule (section)

{{Short description|Formula for the derivative of an inverse function}}
{{about|the computation of the derivative of an invertible function|a condition on which a function is invertible|Inverse function theorem}}
{{refimprove|date=January 2022}}
[[File:Umkehrregel 2.png|thumb|right|250px|The thick blue curve and the thick red curve are inverse to each other. A thin curve is the derivative of the same colored thick curve.

Inverse function 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>]]
{{calculus|expanded=differential}}
In [[calculus]], the '''inverse function rule''' is a [[formula]] that expresses the [[derivative]] of the [[inverse function|inverse]] of a [[bijective]] and [[differentiable function]] {{Mvar|f}} in terms of the derivative of {{Mvar|f}}. More precisely, if 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>, then the inverse function rule is, in [[Lagrange's notation]],

:<math>\left[f^{-1}\right]'(y)=\frac{1}{f'\left( f^{-1}(y) \right)}</math>.

This formula holds in general whenever <math>f</math> is [[continuous function|continuous]] and [[Injective function|injective]] on an interval {{Mvar|I}}, with <math>f</math> being differentiable at <math>f^{-1}(y)</math>(<math>\in I</math>) and where<math>f'(f^{-1}(y)) \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 [[graph of a function|graphs]] that are [[Reflection (mathematics)|reflection]]s, in the line <math>y=x</math>. This reflection operation turns the [[slope|gradient]] of any line into its [[Multiplicative inverse|reciprocal]].<ref>{{Cite web|url=https://oregonstate.edu/instruct/mth251/cq/Stage6/Lesson/inverseDeriv.html|title=Derivatives of Inverse Functions|website=oregonstate.edu|access-date=2019-07-26 |archive-url=https://web.archive.org/web/20210410154136/https://oregonstate.edu/instruct/mth251/cq/Stage6/Lesson/inverseDeriv.html |archive-date=2021-04-10 |url-status=dead}}</ref>

Assuming that <math>f</math> has an inverse in a [[neighborhood (mathematics)|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.

The inverse function rule may also be expressed in [[Leibniz's notation]]. As that notation suggests,

:<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 {{Mvar|x}} and applying the [[chain rule]], yielding that:

:<math>\frac{dx}{dy}\,\cdot\, \frac{dy}{dx} = \frac{dx}{dx}</math>

considering that the derivative of {{Mvar|x}} with respect to ''{{Mvar|x}}'' is 1.