Derivative of Real Area Hyperbolic Cotangent of x over a

Theorem

 * $\dfrac {\mathrm d \left({\coth^{-1} \left({\frac x a}\right)}\right)} {\mathrm d x} = \dfrac {-a} {x^2 - a^2}$

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

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

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


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