Differentiate w.r.t x:

Formula:

$\frac{\mathrm{d}\left(\mathrm{e}^{\mathrm{x}}\right)}{\mathrm{dx}}=\mathrm{e}^{\mathrm{x}}, \frac{\mathrm{d}\left(\mathrm{x}^{\mathrm{n}}\right)}{\mathrm{dx}}=\mathrm{n} \times \mathrm{x}^{\mathrm{n}-1}$ and $\frac{\mathrm{d}(\operatorname{cosec} \mathrm{x})}{\mathrm{dx}}=-\operatorname{cosec} \mathrm{x} \cot \mathrm{x}$

According to the quotient rule of differentiation

if $y=\frac{u}{v}$

$\mathrm{dy} / \mathrm{dx}=\frac{\mathrm{v} \times \frac{\mathrm{du}}{\mathrm{dx}}-\mathrm{u} \times \frac{\mathrm{dv}}{\mathrm{dx}}}{\mathrm{v}^{2}}$

$=\frac{(\operatorname{cosec} 2 x) \times\left(2 e^{2 x}+3 x^{2}\right)-\left(e^{2 x}+x^{3}\right) \times(-2 \operatorname{cosec} 2 x \cot 2 x)}{(\operatorname{cosec} 2 x)^{2}}$

$=\frac{2 e^{2 x} \operatorname{cosec} 2 x+3 x^{2} \operatorname{cosec} 2 x+2 e^{2 x} \operatorname{cosec} 2 x \cot 2 x+2 x^{3} \operatorname{cosec} 2 x \cot 2 x}{(\operatorname{cosec} 2 x)^{2}}$

$=\frac{2 e^{2 x} \operatorname{cosec} 2 x(1+\cot 2 x)+3 x^{2} \operatorname{cosec} 2 x(1+\cot 2 x)}{(\operatorname{cosec} 2 x)^{2}}$

$=\frac{(1+\cot 2 x)\left(2 e^{x} \operatorname{cosec} 2 x+3 x^{2} \operatorname{cosec} 2 x\right)}{(\operatorname{cosec} 2 x)^{2}}$

$=\frac{(1+\cot 2 x)\left(2 e^{x}+3 x^{2}\right)(\operatorname{cosec} 2 x)}{(\operatorname{cosec} 2 x)^{2}}$

$=\frac{(1+\cot 2 x)\left(2 e^{x}+3 x^{2}\right)}{(\operatorname{cosec} 2 x)^{1}}$

$=(1+\cot 2 x)\left(2 e^{x}+3 x^{2}\right)(\sin 2 x)$