# How do you differentiate f(x)=sqrt(1/csc(2/x )  using the chain rule?

$f ' \left(x\right) = - \csc \frac{\frac{2}{x}}{{x}^{2} \cdot \sqrt{\sin \left(\frac{2}{x}\right)}}$
Note that $\csc \left(x\right) = \frac{1}{\sin} \left(x\right)$
$f ' \left(x\right) = \frac{1}{2} \cdot {\left(\frac{1}{\csc} \left(\frac{2}{x}\right)\right)}^{- \frac{1}{2}} \cdot \csc \left(\frac{x}{2}\right) \cdot \left(- \frac{2}{x} ^ 2\right)$