Answer
${dy}={{x}^{(\frac{-3}{2})}}\csc^2({{x}^{(\frac{-1}{2})}}){dx}$
Work Step by Step
We evaluate the function: $y=2\cot(\frac{1}{\sqrt{x}})$
$y=2\cot({{x}^{\frac{-1}{2}}})$
On differentiating the above:
$\frac{dy}{dx}=\frac{d(2{\cot{x^\frac{-1}{2}}}))}{dx}$
or $\frac{dy}{dx}=-2\csc^2({{x}^{(\frac{-1}{2})}})\frac{d(({{{x}^{(\frac{-1}{2})}}})}{dx}$
or $\frac{dy}{dx}={{x}^{(\frac{-3}{2})}}\csc^2({{x}^{(\frac{-1}{2})}})$
or ${dy}={{x}^{(\frac{-3}{2})}}\csc^2({{x}^{(\frac{-1}{2})}}){dx}$
The final answer is: ${dy}={{x}^{(\frac{-3}{2})}}\csc^2({{x}^{(\frac{-1}{2})}}){dx}$
