Differentiate w.r.t x

Let $y=e^{-5 x} \cot 4 x, z=e^{-5 x}$ and $w=\cot 4 x$

Formula

$\frac{\mathrm{d}\left(\mathrm{e}^{\mathrm{x}}\right)}{\mathrm{dx}}=\mathrm{e}^{\mathrm{x}}$ and $\frac{\mathrm{d}(\cot \mathrm{x})}{\mathrm{dx}}=-\operatorname{cosec}^{2} \mathrm{x}$

According to the product rule of differentiation

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

$=\left[\cot 4 x \times\left(-5 e^{-5 x}\right)\right]+\left[e^{-5 x} \times\left(-4 \operatorname{cosec}^{2} 4 x\right)\right]$

$=-e^{-5 x} \times\left[5 \cot 4 x+4 \operatorname{cosec}^{2} 4 x\right]$