Differentiate the functions with respect to x.

$\frac{d}{d x}\left[2 \sqrt{\cot \left(x^{2}\right)}\right]$

$=2 \cdot \frac{1}{2 \sqrt{\cot \left(x^{2}\right)}} \times \frac{d}{d x}\left[\cot \left(x^{2}\right)\right]$

$=\sqrt{\frac{\sin \left(x^{2}\right)}{\cos \left(x^{2}\right)}} \times-\operatorname{cosec}^{2}\left(x^{2}\right) \times \frac{d}{d x}\left(x^{2}\right)$

$=-\sqrt{\frac{\sin \left(x^{2}\right)}{\cos \left(x^{2}\right)}} \times \frac{1}{\sin ^{2}\left(x^{2}\right)} \times(2 x)$

$=\frac{-2 x}{\sqrt{\cos x^{2}} \sqrt{\sin x^{2}} \sin x^{2}}$

$=\frac{-2 \sqrt{2} x}{\sqrt{2 \sin x^{2} \cos x^{2}} \sin x^{2}}$

$=\frac{-2 \sqrt{2} x}{\sin x^{2} \sqrt{\sin 2 x^{2}}}$