Derivative of Real Area Hyperbolic Tangent of x over a

Theorem

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

where $-a < x < a$.

Proof
Let $-a < x < a$.

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

$\tanh^{-1} \dfrac x a$ is not defined when either $x \le -a$ or $x \ge a$.

Also see

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


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


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


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


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