Derivatives of Inverse Trigonometric Functions

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Theorem

This page gathers together derivatives of inverse trigonometric functions.


Derivative of Arcsine Function

$\dfrac {\map \d {\arcsin x} } {\d x} = \dfrac 1 {\sqrt {1 - x^2} }$


Derivative of Arccosine Function

$\map {D_x} {\arccos x} = \dfrac {-1} {\sqrt {1 - x^2} }$


Derivative of Arctangent Function

$\dfrac {\map \d {\arctan x} } {\d x} = \dfrac 1 {1 + x^2}$


Derivative of Arccotangent Function

$\dfrac {\map \d {\arccot x} } {\d x} = \dfrac {-1} {1 + x^2}$


Derivative of Arcsecant Function

$\dfrac {\map \d {\arcsec x} } {\d x} = \dfrac 1 {\size x \sqrt {x^2 - 1} } = \begin {cases} \dfrac {+1} {x \sqrt {x^2 - 1} } & : 0 < \arcsec x < \dfrac \pi 2 \ (\text {that is: $x > 1$}) \\ \dfrac {-1} {x \sqrt {x^2 - 1} } & : \dfrac \pi 2 < \arcsec x < \pi \ (\text {that is: $x < -1$}) \\ \end{cases}$


Derivative of Arccosecant Function

$\dfrac {\map \d {\arccsc x} } {\d x} = \dfrac {-1} {\size x \sqrt {x^2 - 1} } = \begin {cases} \dfrac {-1} {x \sqrt {x^2 - 1} } & : 0 < \arccsc x < \dfrac \pi 2 \ (\text {that is: $x > 1$}) \\ \dfrac {+1} {x \sqrt {x^2 - 1} } & : -\dfrac \pi 2 < \arccsc x < 0 \ (\text {that is: $x < -1$}) \\ \end{cases}$