Derivative of Inverse Hyperbolic Sine

Theorem
Let $x \in \R$ be a real number.

Let $\arsinh x$ denote the inverse hyperbolic sine of $x$.

Then:
 * $\map {\dfrac \d {\d x} } {\arsinh x} = \dfrac 1 {\sqrt {x^2 + 1} }$

Proof
For all $x \in \R$ we have that $\cosh y \ge 1$.

Thus it follows that it is necessary to take the positive root of $\sqrt {\sinh^2 y + 1}$.

So: