The derivative of e3x sin4x with respect to x, is

# The derivative of  with respect to $\mathrm{x},$ is

1. A

$5{\mathrm{e}}^{3\mathrm{x}}\mathrm{sin}\left(4\mathrm{x}+{\mathrm{Tan}}^{-1}\frac{4}{3}\right)$

2. B

$5{\mathrm{e}}^{3\mathrm{x}}\mathrm{sin}\left(4\mathrm{x}-{\mathrm{Tan}}^{-1}\frac{4}{3}\right)$

3. C

$5{\mathrm{e}}^{3\mathrm{x}}\mathrm{sin}\left(4\mathrm{x}+{\mathrm{Tan}}^{-1}\frac{3}{4}\right)$

4. D

$5{\mathrm{e}}^{3\mathrm{x}}\mathrm{sin}\left(4\mathrm{x}-{\mathrm{Tan}}^{-1}\frac{3}{4}\right)$

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### Solution:

Suppose that $y={e}^{3x}\mathrm{sin}4x$

Differentiate both sides

Here $\mathrm{sin}A=\frac{4}{5},\mathrm{cos}A=\frac{3}{5}⇒\mathrm{tan}A=\frac{4}{3}$

Therefore, $\frac{dy}{dx}=5{e}^{3x}\left(\mathrm{sin}\left(4x+{\mathrm{tan}}^{-1}\left(\frac{4}{3}\right)\right)\right)$  +91

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