y=exlogx.sinx then dydx=

# $y={e}^{x}\mathrm{log}x.\mathrm{sin}x\text{\hspace{0.17em}}then\text{\hspace{0.17em}}\frac{dy}{dx}=$

1. A

${e}^{x}\left(\mathrm{log}x\mathrm{cos}x+\frac{\mathrm{sin}x}{x}+\mathrm{log}x\mathrm{sin}x\right)$

2. B

${e}^{x}\left(\mathrm{log}x\mathrm{cos}x-\frac{\mathrm{sin}x}{x}+\mathrm{log}x\mathrm{sin}x\right)$

3. C

${e}^{x}\left(\mathrm{log}x+\frac{\mathrm{sin}x}{x}+\mathrm{log}x\mathrm{sin}x\right)$

4. D

all the above

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### Solution:

$\begin{array}{l}y={e}^{x}\mathrm{log}x\mathrm{sin}x\\ {y}^{1}={e}^{x}\mathrm{log}x\frac{d}{dx}\left(\mathrm{sin}x\right)+{e}^{x}\mathrm{sin}x\frac{d}{dx}\left(\mathrm{log}x\right)+\mathrm{log}x.\mathrm{sin}x\frac{d}{dx}\left({e}^{x}\right)\\ {y}^{1}={e}^{x}\mathrm{log}x\mathrm{cos}x+\frac{{e}^{x}\mathrm{sin}x}{x}+{e}^{x}\mathrm{log}x\mathrm{sin}x\\ \frac{dy}{dx}={e}^{x}\left(\mathrm{log}x\mathrm{cos}x+\frac{\mathrm{sin}x}{x}+\mathrm{log}x\mathrm{sin}x\right)\end{array}$

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