Derivative of Real Area Hyperbolic Tangent of x over a

Theorem

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

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 presented as
Some sources present this as:


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

Also see

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


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


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


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


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