Correct Answer - Option 1 : 1
Let
\(y = 2 \sqrt{cot(x^2)}\)
\(\dfrac{dy}{dx} = 2 \dfrac{d}{dx} ( \sqrt{cot x^2)}\)
\(= 2. \dfrac{1}{2 \sqrt{ cot x^2}.}\dfrac{d}{dx} ( {cot x^2)}\)
\(= \dfrac{1}{\sqrt{ cot x^2}} [-cosec^2 (x^2)] \dfrac{d}{dx}(x^2)\)
\(= \dfrac{ -cosec^2 (x^2)}{\sqrt { cot x^2}}.(2x)= - \dfrac{2x \sqrt { tan x^2}}{sin^2 (x^2)}\)
\(= \dfrac{-2x \sqrt {sin x^2 }}{sin^2 (x^2) \sqrt {cos x^2}} = \dfrac{-2x}{sin(x^2) \sqrt { sinx^2 cos x^2 }}\)
\(= \dfrac{ -2 \sqrt 2 x}{ sin^2 (x^2) \sqrt { 2 sin x^2 cos x^2 }}= \dfrac{-2 \sqrt 2 x}{ sin(x^2) \sqrt { sin (2x^2)}}\)