Derivative of Arcsecant Function/Corollary 1

Corollary to Derivative of Arcsecant Function
Let $x \in \R$.

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

Then:
 * $\dfrac {\map \d {\map \arcsec {\frac x a} } } {\d x} = \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
Similarly:

Also see

 * Derivative of $\arcsin \dfrac x a$


 * Derivative of $\arccos \dfrac x a$


 * Derivative of $\arctan \dfrac x a$


 * Derivative of $\arccot \dfrac x a$


 * Derivative of $\arccsc \dfrac x a$