Answer
$\frac{1}{{x}\sqrt ({x^2}-{1})}$
Work Step by Step
Given that $y$ = $cos^{-1}(\frac{1}{x})$
on substituting $\frac{1}{x}$ = $u$
$\frac{dy}{dx}$ = $\frac{{dcos^{-1}{u}}}{du}$ $\times$ $\frac{du}{dx}$
or $\frac{dy}{dx}$ = $\frac{1}{\sqrt (1-\frac{1}{x^2})}$ $\times$ $(-\frac{1}{x^2})$
so $\frac{dy}{dx}$ = $\frac{1}{{x^2}\sqrt (1-\frac{1}{x^2})}$
or $\frac{dy}{dx}$ = $\frac{1}{{x}\sqrt ({x^2}-{1})}$
The final answer is: $\frac{1}{{x}\sqrt ({x^2}-{1})}$