How do you differentiate f(x)=xe^(2x)sinx using the product rule?

$f ' \left(x\right) = {e}^{2 x} \left(2 x \setminus \sin x + x \setminus \cos x + \setminus \sin x\right)$

Explanation:

Given function:

$f \left(x\right) = x {e}^{2 x} \setminus \sin x$

differentiating above function using product rule as follows

$\frac{d}{\mathrm{dx}} f \left(x\right) = \frac{d}{\mathrm{dx}} \left(x {e}^{2 x} \setminus \sin x\right)$

$f ' \left(x\right) = x {e}^{2 x} \frac{d}{\mathrm{dx}} \left(\setminus \sin x\right) + x \setminus \sin x \frac{d}{\mathrm{dx}} \left({e}^{2 x}\right) + {e}^{2 x} \setminus \sin x \frac{d}{\mathrm{dx}} \left(x\right)$

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

$= x {e}^{2 x} \setminus \cos x + 2 x {e}^{2 x} \setminus \sin x + {e}^{2 x} \setminus \sin x$

$= {e}^{2 x} \left(2 x \setminus \sin x + x \setminus \cos x + \setminus \sin x\right)$