Derivative of Arccosecant Function/Corollary

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

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

Then:
 * $\dfrac {\mathrm d \left({\operatorname{arccsc} \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{arccsc} \dfrac x a < \dfrac \pi 2 \\

\dfrac a {x \sqrt {x^2 - a^2} } & : -\dfrac \pi 2 < \operatorname{arccsc} \dfrac x a < 0 \\ \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 $\operatorname{arcsec} \dfrac x a$