Differentiate the function with respect to $x$: $2 \sqrt{\cot \left(x^{2}\right)}$

Let $y = 2 \sqrt{\cot \left(x^{2}\right)}$.
Applying the chain rule,we differentiate with respect to $x$:
$\frac{dy}{dx} = 2 \cdot \frac{1}{2\sqrt{\cot(x^2)}} \cdot \frac{d}{dx}[\cot(x^2)]$
$= \frac{1}{\sqrt{\cot(x^2)}} \cdot [-\csc^2(x^2) \cdot \frac{d}{dx}(x^2)]$
$= \frac{1}{\sqrt{\frac{\cos(x^2)}{\sin(x^2)}}} \cdot [-\frac{1}{\sin^2(x^2)} \cdot 2x]$
$= -\sqrt{\frac{\sin(x^2)}{\cos(x^2)}} \cdot \frac{2x}{\sin^2(x^2)}$
$= -\frac{2x}{\sqrt{\cos(x^2) \sin(x^2) \sin(x^2)}}$
$= -\frac{2x}{\sin(x^2) \sqrt{\sin(x^2)\cos(x^2)}}$
Multiplying numerator and denominator by $\sqrt{2}$:
$= -\frac{2\sqrt{2}x}{\sin(x^2) \sqrt{2\sin(x^2)\cos(x^2)}}$
$= -\frac{2\sqrt{2}x}{\sin(x^2) \sqrt{\sin(2x^2)}}$