derivative of inverse function

Theorem.  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

$$f_{\leftarrow }^{\prime }(y)=\frac{1}{f^{\prime }(x)}.$$ (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.

{it 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

$$f^{\prime }(f_{\leftarrow }(y))\cdot f_{\leftarrow }^{\prime }(y)=1,$$

whence

$$f_{\leftarrow }^{\prime }(y)=\frac{1}{f^{\prime }(f_{\leftarrow }(y))}.$$

This is same as the asserted (1).

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. $y=x^2$,   $x=\pm \sqrt{y}$;  $\frac{dx}{dy}=\frac{1}{\frac{dy}{dx}}=\frac{1}{2x}=\frac{1}{\pm 2\sqrt{y}}$

If the variable symbol $y$ in those results is changed to $x$, the results can be written

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

Title	derivative of inverse function
Canonical name	DerivativeOfInverseFunction
Date of creation	2015-02-21 16:02:46
Last modified on	2015-02-21 16:02:46
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	12
Author	pahio (2872)
Entry type	Theorem
Classification	msc 26A24
Related topic	InverseFunctionTheorem
Related topic	Derivative2
Related topic	DerivativeOfTheNaturalLogarithmFunction
Related topic	CyclometricFunctions
Related topic	SquareRoot
Related topic	LimitExamples
Related topic	IntegrationOfSqrtx21