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derivative of inverse function
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(Theorem)
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Theorem 1 If the real function $f$ has an inverse function $f_\leftarrow$ and the derivative of $f$ at the point $x = f_\leftarrow(y)$ is distinct from zero, then $f_\leftarrow$ is also differentiable at the point $y$ and \begin{equation} \label{d} f_\leftarrow'(y) =
\frac{1}{f'(x)}. \end{equation}That is, the derivatives of a function and its inverse function are inverse numbers of each other, provided that they have been taken at the points which correspond to each other.
Proof. Now we have $$f(f_\leftarrow(y)) = f(x) =y.$$ The derivatives of both sides must be equal: $$\frac{d}{dy}\left[f(f_\leftarrow(y))\right] = \frac{d}{dy}y$$ Using the chain rule we get $$f'(f_\leftarrow(y))\cdot f_\leftarrow'(y) = 1,$$ whence $$f_\leftarrow'(y) = \frac{1}{f'(f_\leftarrow(y))}.$$ This is same as the asserted equation ( ![[*]](http://images.planetmath.org:8080/cache/objects/9359/js//usr/share/latex2html/icons/crossref.png "[*]") ).
Examples. For simplicity, we express here the functions by symbols $y$ and the inverse functions by $x$ .
- $y = \tan{x}$ , $x = \arctan{y}$ ; $\frac{dx}{dy} = \frac{1}{\frac{dy}{dx}} = \frac{1}{1+\tan^2{x}} = \frac{1}{1+y^2}$
- $y = \sin{x}$ , $x = \arcsin{y}$ ; $\frac{dx}{dy} = \frac{1}{\frac{dy}{dx}} = \frac{1}{\cos{x}} = \frac{1}{+\sqrt{1-\sin^2{x}}} = +\frac{1}{\sqrt{1-y^2}}$
- $y = x^2$ , $x = \pm\sqrt{y}$ ; $\frac{dx}{dy} = \frac{1}{\frac{dy}{dx}} = \frac{1}{2x} = \frac{1}{\pm2\sqrt{y}}$
If the variable symbol $y$ in those results is changed to $x$ , the results can be written $$\frac{d}{dx}\arctan{x} = \frac{1}{1+x^2},\quad \frac{d}{dx}\arcsin{x} = \frac{1}{\sqrt{1-x^2}},\quad \frac{d}{dx}\sqrt{x} = \frac{1}{2\sqrt{x}}.$$
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"derivative of inverse function" is owned by pahio. [ full author list (2) ]
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Cross-references: variable, chain rule, sides, inverse numbers, function, differentiable, point, derivative, inverse function, real function
There are 3 references to this entry.
This is version 8 of derivative of inverse function, born on 2007-05-10, modified 2007-10-16.
Object id is 9359, canonical name is DerivativeOfInverseFunction.
Accessed 8828 times total.
Classification:
| AMS MSC: | 26A24 (Real functions :: Functions of one variable :: Differentiation : general theory, generalized derivatives, mean-value theorems) |
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Pending Errata and Addenda
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