ddx(xn⋅sinx⋅ex)=

# $\frac{d}{dx}\left({x}^{n}\cdot \mathrm{sin}x\cdot {e}^{x}\right)=$

1. A

${e}^{x}\left[{x}^{n}\cdot \mathrm{sin}x+{x}^{3}\cdot \mathrm{cos}x-{x}^{n-1}\cdot \mathrm{sin}x\right]$

2. B

${e}^{x}\left[{x}^{n}\cdot \mathrm{sin}x+{x}^{3}\cdot \mathrm{cos}x+{x}^{n-1}\cdot \mathrm{sin}x\right]$

3. C

${e}^{x}\left[{x}^{n}\cdot \mathrm{sin}x-{x}^{3}\cdot \mathrm{cos}x+{x}^{n-1}\cdot n\mathrm{sin}x\right]$

4. D

None of these

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### Solution:

$\frac{d}{dx}\left({x}^{n}\cdot \mathrm{sin}x\cdot {e}^{x}\right)={x}^{n}\cdot \mathrm{sin}x\cdot {e}^{x}+{x}^{n}\cdot {e}^{x}\cdot \mathrm{cos}x+\mathrm{sin}x\cdot {e}^{x}\cdot n.{x}^{n-1}$

$={e}^{x}\left[{x}^{n}\cdot \mathrm{sin}x+{x}^{n}\cdot \mathrm{cos}x+{x}^{n-1}\cdot n\mathrm{sin}x\right]$

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