Question

Differentiate the following function with respect to $x$:
$\sin h^{-1}\left(\sqrt{x}\right)$

Answer 1

$sin{h}^{-1}x=ln\left(x+\sqrt{x^2+1}\right)$

$\Rightarrow sin{h}^{-1}\sqrt{x}=ln\left(\sqrt{x}+\sqrt{x+1}\right)$

$\Rightarrow \frac{d}{dx}sin{h}^{-1}\sqrt{x}=\frac{1}{\sqrt{x}+\sqrt{x+1}}\times [\frac{1}{2\sqrt{x}}+\frac{1}{2\sqrt{x+1}}]$

$\Rightarrow \frac{d}{dx}sin{h}^{-1}\sqrt{x}=\frac{1}{2\sqrt{x(x+1)}}$