Derivative of Real Area Hyperbolic Cosine of x over a

Theorem

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

where $x > a$.

Proof
Let $x > a$.

Then $\dfrac x a > 1$ and so:

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

Also see

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


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


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


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


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