Question

Differentiate the expressions with respect to $x$
$\cos h^{-1}\left(\frac{1}{x}\right)(x>0)$

Answer 1

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

$\Rightarrow cos{h}^{-1}\left(\frac{1}{x}\right)=ln\left(\left(1/x\right)+\sqrt{(1/x{)}^2-1}\right)$

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

$\Rightarrow \frac{d}{dx}cos{h}^{-1}\left(\frac{1}{x}\right)==\frac{x}{1+\sqrt{1-x^2}}\left[\frac{-1}{x^2}\right(\frac{1+\sqrt{1-x^2}}{\sqrt{1-x^2}}\left)\right]$

$\Rightarrow \frac{d}{dx}cos{h}^{-1}\left(\frac{1}{x}\right)=\frac{-1}{x\left(\sqrt{1-x^2}\right)}$