Find the value

Let $y=e^{x} \log (\sin 2 x), z=e^{x}$ and $w=\log (\sin 2 x)$

Formula :

$\frac{\mathrm{d}\left(\mathrm{e}^{\mathrm{x}}\right)}{\mathrm{dx}}=\mathrm{e}^{\mathrm{x}}, \frac{\mathrm{d}(\log \mathrm{x})}{\mathrm{dx}}=\frac{1}{\mathrm{x}}$ and $\frac{\mathrm{d}(\sin \mathrm{x})}{\mathrm{dx}}=\cos \mathrm{x}$

According to the product rule of differentiation

$\mathrm{dy} / \mathrm{dx}=\mathrm{w} \times \frac{\mathrm{dz}}{\mathrm{dx}}+\mathrm{z} \times \frac{\mathrm{dw}}{\mathrm{dx}}$

$=\left[\log (\sin 2 x) \times\left(e^{x}\right)\right]+\left[e^{x} \times \frac{1}{\sin 2 x} \times 2 \cos 2 x\right]$

$=e^{x} \times\left[\log (\sin 2 x)+\frac{2 \cos 2 x}{\sin 2 x}\right]$

$=e^{x} \times[\log (\sin 2 x)+2 \cot 2 x]$