Ableitungsregel für Areasinus hyperbolicus

\begin{align*} {\Bigl[ \arsinh(x) \Bigr]}' &= {\left[ \ln\left( x + \sqrt{x^2 + 1} \right) \right]}' \\[0.75em] &= \frac{1}{x + \sqrt{x^2 + 1}} \cdot {\left[ x + \sqrt{x^2 + 1} \right]}' \\[0.75em] &= \frac{1}{x + \sqrt{x^2 + 1}} \cdot \left( 1 + \frac{{\left[ x^2 + 1\right]}'}{2 \cdot \sqrt{x^2+1}} \right) \\[0.75em] &= \frac{1}{x + \sqrt{x^2 + 1}} \cdot \left( 1 + \frac{2x}{2 \cdot \sqrt{x^2+1}} \right) \\[0.75em] &= \frac{1}{x + \sqrt{x^2 + 1}} \cdot \frac{\sqrt{x^2 + 1} + x}{\sqrt{x^2+1}} \\[0.75em] &= \frac{1}{\sqrt{x^2 + 1}} \end{align*}