Derivative of Real Area Hyperbolic Cotangent of x over a

Theorem

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

where either $x < -a$ or $x > a$.

Proof
Let either $x < -a$ or $x > a$.

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

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

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