# derivative of inverse function

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. 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. 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. 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

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