Derivative of Real Area Hyperbolic Cosecant of x over a

Theorem

 * $\dfrac {\map \d {\csch^{-1} } {\frac x a} } {\d x} = \dfrac {-a} {\size x \sqrt {a^2 + x^2} }$

where $x \ne 0$.

Proof
Let $0 < x < a$.

Then $0 < \dfrac x a < 1$ and so:

$\csch^{-1} \dfrac x a$ is not defined when $x = 0$.

Also see

 * Derivative of $\sinh^{-1} \dfrac x a$


 * Derivative of $\cosh^{-1} \dfrac x a$


 * Derivative of $\tanh^{-1} \dfrac x a$


 * Derivative of $\coth^{-1} \dfrac x a$


 * Derivative of $\sech^{-1} \dfrac x a$