Derivative of Real Area Hyperbolic Cosine of x over a

Theorem

 * $\dfrac {\map \d {\map \arcosh {\frac x a} } } {\d x} = \dfrac 1 {\sqrt {x^2 - a^2} }$

where $x > a$.

Proof
Let $x > a$.

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

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

When $x = a$ we have that $\sqrt {x^2 - a^2} = 0$ and so $\dfrac 1 {\sqrt {x^2 - a^2} }$ is not defined.

Hence the restriction on the domain.

Also see

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


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


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


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


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