Question

$2 \sqrt{ \cot( {x}^{2} ) }$
Differential the function with repect to x..​

Answer 1

♣ Qᴜᴇꜱᴛɪᴏɴ :

$\boxed{\bf{\dfrac{d}{dx}\left(2\sqrt{cot\left(x^2\right)}\right)}}$

♣ ᴀɴꜱᴡᴇʀ :

$\mathrm{Take\:the\:constant\:out}:\quad \left(a\cdot f\right)'=a\cdot f\:'$

$=2\dfrac{d}{dx}\left(\sqrt{\cot \left(x^2\right)}\right)$

$\sf{Apply\:\:the\:\:chain\:\:rule\:\::\dfrac{1}{2\sqrt{\cot \left(x^2\right)}}\dfrac{d}{dx}\left(\cot \left(x^2\right)\right)}$

$\sf{=\dfrac{1}{2\sqrt{\cot \left(x^2\right)}}\dfrac{d}{dx}\left(\cot \left(x^2\right)\right)}$

$\sf{\dfrac{d}{dx}\left(\cot \left(x^2\right)\right) = -\csc ^2\left(x^2\right)\cdot \:2x}$

$=2\cdot \dfrac{1}{2\sqrt{\cot \left(x^2\right)}}\left(-\csc ^2\left(x^2\right)\cdot \:2x\right)$

$\bigstar \bf{Simplify\:\::2\cdot \dfrac{1}{2\sqrt{\cot \left(x^2\right)}}\left(-\csc ^2\left(x^2\right)\cdot \:2x\right)}$

$\boxed{\bf{=-\dfrac{2x\csc ^2\left(x^2\right)}{\sqrt{\cot \left(x^2\right)}}}}$