Derivative of Real Area Hyperbolic Cotangent of x over a

Theorem

 * $\map {\dfrac \d {\d x} } {\map {\coth^{-1} } {\dfrac x a} } = \dfrac {-a} {x^2 - a^2}$

where $x^2 > a^2$.

Proof
Let $x^2 > a^2$.

Then either $\dfrac x a < -1$ or $\dfrac x a > 1$ and so:

$\coth^{-1} \dfrac x a$ is not defined when $x^2 \le a^2$.

Also presented as
This result can also be (and usually is) reported as:
 * $\map {\dfrac {\d} {\d x} } {\map {\coth^{-1} } {\dfrac x a} } = \dfrac a {a^2 - x^2}$

but this obscures the fact that $x^2 > a^2$ in order for it to be defined.

Some sources present it as:


 * $\map {\dfrac {\d} {\d x} } {\dfrac 1 a \map {\coth^{-1} } {\dfrac x a} } = \dfrac 1 {a^2 - x^2}$

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


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