Derivative of Real Area Hyperbolic Cosecant of x over a

Theorem

 * $\dfrac {\mathrm d \left({\operatorname{csch}^{-1} \left({\frac x a}\right)}\right)} {\mathrm d x} = \dfrac {-a} {\left\vert{x}\right\vert \sqrt{a^2 + x^2} }$

where $x \ne 0$.

Proof
Let $0 < x < a$.

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

$\operatorname{sech}^{-1} \dfrac x a$ 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 $\operatorname{sech}^{-1} \dfrac x a$