ddx[ex.sinx.ax]=

# $\frac{d}{dx}\left[{e}^{x}.\mathrm{sin}x.{a}^{x}\right]=$

1. A

${e}^{x}{a}^{x}\left[\mathrm{sin}x.\mathrm{log}a-\mathrm{cos}x+\mathrm{sin}x\right]$

2. B

${e}^{x}{a}^{x}\left[\mathrm{sin}x.\mathrm{log}a-\mathrm{cos}x-\mathrm{sin}x\right]$

3. C

${e}^{x}{a}^{x}\left[\mathrm{sin}x.\mathrm{log}a+\mathrm{cos}x-\mathrm{sin}x\right]$

4. D

${e}^{x}{a}^{x}\left[\mathrm{sin}x.\mathrm{log}a+\mathrm{cos}x+\mathrm{sin}x\right]$

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### Solution:

$\begin{array}{l}Here\text{\hspace{0.17em}}u={e}^{x},v=\mathrm{sin}x,w={a}^{x}\\ By\text{\hspace{0.17em}using\hspace{0.17em}}\frac{d}{dx}\left[uvw\right]=uv\frac{d}{dx}\left(w\right)+vw\frac{d}{dx}\left(u\right)+uw\frac{d}{dx}\left(v\right)\\ \frac{d}{dx}\left[{e}^{x}.\mathrm{sin}x.{a}^{x}\right]={e}^{x}\mathrm{sin}x\frac{d}{dx}\left({a}^{x}\right)+{e}^{x}.{a}^{x}\frac{d}{dx}\left(\mathrm{sin}x\right)+{a}^{x}\mathrm{sin}x\frac{d}{dx}\left({e}^{x}\right)\\ ={e}^{x}\mathrm{sin}x\left({a}^{x}.\mathrm{log}a\right)+{e}^{x}{a}^{x}\mathrm{cos}x+{a}^{x}\mathrm{sin}x\left({e}^{x}\right)\\ ={e}^{x}{a}^{x}\left[\mathrm{sin}x.\mathrm{log}a+\mathrm{cos}x+\mathrm{sin}x\right]\end{array}$

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