Derivative of Arccosecant Function/Corollary

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

Let $\arccsc \dfrac x a$ be the arccosecant of $\dfrac x a$.

Then:
 * $\dfrac {\map \d {\arccsc \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 < \arccsc \dfrac x a < \dfrac \pi 2 \\

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


 * Derivative of $\arcsec \dfrac x a$