Derivative of Arcsecant Function/Corollary 1

Corollary to Derivative of Arcsecant Function

Let $x \in \R$.

Let $\arcsec \dfrac x a$ be the arcsecant of $\dfrac x a$.

Then:

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

Proof

$\ds \map {\dfrac \d {\d x} } {\map \arcsec {\dfrac x a} }$

$=$

$\ds \frac 1 a \frac 1 {\size {\frac x a} \sqrt {\paren {\frac x a}^2 - 1} }$

Derivative of Arcsecant Function and Derivative of Function of Constant Multiple

$\ds $

$=$

$\ds \frac 1 a \frac 1 {\size {\frac x a} \frac {\sqrt {x^2 - a^2} } a}$

$\ds $

$=$

$\ds \frac 1 a \frac {a^2} {\size x {\sqrt {x^2 - a^2} } }$

$\ds $

$=$

$\ds \frac a {\size x {\sqrt {x^2 - a^2} } }$

$\Box$

Similarly:

$\ds \map {\dfrac \d {\d x} } {\map \arcsec {\dfrac x a} }$

$=$

$\ds \begin {cases} \dfrac 1 a \dfrac {+1} {\frac x a \sqrt {\paren {\frac x a}^2 - 1} } & : 0 < \arcsec \dfrac x a < \dfrac \pi 2 \ (\text {that is: $\dfrac x a > 1$}) \\ \dfrac 1 a \dfrac {-1} {\frac x a \sqrt {\paren {\frac x a}^2 - 1} } & : \dfrac \pi 2 < \arcsec \dfrac x a < \pi \ (\text {that is: $\dfrac x a < -1$}) \\ \end{cases}$

Derivative of Arcsecant Function
and Derivative of Function of Constant Multiple

$\ds $

$=$

$\ds \begin {cases} \dfrac a {x \sqrt {x^2 - a^2} } & : 0 < \arcsec \dfrac x a < \dfrac \pi 2 \\ \dfrac {-a} {x \sqrt {x^2 - a^2} } & : \dfrac \pi 2 < \arcsec \dfrac x a < \pi \\ \end{cases}$

simplifying as above

$\blacksquare$