Question

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

• A. $x{e}^x\left(2\sin \left(x\right)+x\sin \left(x\right)+x\cos \left(x\right)\right)$
• B. $x{e}^x\left(2\sin \left(x\right)+x\sin \left(x\right)-\cos \left(x\right)\right)$
• C. $x{e}^x\left(2\sin \left(x\right)+x\sin \left(x\right)+\cos \left(x\right)\right)$
• D. none of these

Answer 1

The correct option is A

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

Explanation for correct option:

Given, $\frac{d}{dx}\left(x^2{e}^x\sin \left(x\right)\right)$

Apply theorem for multiplication.$\frac{d}{dx}\left(u·v\right)=\left(\frac{du}{dx}\times v+u\frac{dv}{dx}\right)$

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

Hence, correct answer is option $\left(A\right)$