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

$\ds \frac {\map \d {\arccsc \frac x a} } {\d x}$

$=$

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

Derivative of Arccosecant 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:

$\ds \frac {\map \d {\arccsc \frac x a} } {\d x}$

$=$

$\ds \begin{cases} \dfrac 1 a \dfrac {-1} {\frac x a \sqrt {\paren {\frac x a}^2 - 1} } & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\ \dfrac 1 a \dfrac {+1} {\frac x a \sqrt {\paren {\frac x a}^2 - 1} } & : -\dfrac \pi 2 < \arccsc \dfrac x a < 0 \\ \end{cases}$

Derivative of Arccosecant Function and
Derivative of Function of Constant Multiple

$\ds $

$=$

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

simplifying as above

$\blacksquare$