Derivative of Arcsecant Function/Corollary 1

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

Let $\operatorname{arcsec} \left({\dfrac x a}\right)$ be the arcsecant of $\dfrac x a$.

Then:
 * $\dfrac {\mathrm d \left({\operatorname{arcsec} \left({\frac x a}\right) }\right)} {\mathrm d x} = \dfrac a {\left\vert{x}\right\vert \sqrt {x^2 - a^2} } = \begin{cases} \dfrac a {x \sqrt {x^2 - a^2} } & : 0 < \operatorname{arcsec} \dfrac x a < \dfrac \pi 2 \\

\dfrac {-a} {x \sqrt {x^2 - a^2} } & : \dfrac \pi 2 < \operatorname{arcsec} \dfrac x a < \pi \\ \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 $\operatorname{arccot} \dfrac x a$


 * Derivative of $\operatorname{arccsc} \dfrac x a$