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[parent] derivative of inverse function (Theorem)
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
$\displaystyle f_\leftarrow'(y) = \frac{1}{f'(x)}.$ (1)

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
$\displaystyle f(f_\leftarrow(y)) = f(x) =y.$
The derivatives of both sides must be equal:
$\displaystyle \frac{d}{dy}\left[f(f_\leftarrow(y))\right] = \frac{d}{dy}y$
Using the chain rule we get
$\displaystyle f'(f_\leftarrow(y))\cdot f_\leftarrow'(y) = 1,$
whence
$\displaystyle f_\leftarrow'(y) = \frac{1}{f'(f_\leftarrow(y))}.$
This is same as the asserted equation (1).
$ \qedsymbol$

Examples. For simplicity, we express here the functions by symbols $ y$ and the inverse functions by $ x$.

  1. $ 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}$
  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}}$
  3. $ 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
$\displaystyle \frac{d}{dx}\arctan{x} = \frac{1}{1+x^2},\quad \frac{d}{dx}\arcsin{x} = \frac{1}{\sqrt{1-x^2}},\quad \frac{d}{dx}\sqrt{x} = \frac{1}{2\sqrt{x}}.$



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See Also: inverse function theorem, derivative, alternative definition of the natural logarithm, cyclometric functions, square root, limit examples


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Cross-references: variable, chain rule, sides, inverse numbers, function, differentiable, point, derivative, inverse function, real function
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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 2489 times total.

Classification:
AMS MSC: 26A24 (Real functions :: Functions of one variable :: Differentiation : general theory, generalized derivatives, mean-value theorems)
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