Differentiate the following functions with respect to x: e^ 3x 2x - Vidyadip

Question

Differentiate the following functions with respect to x:
$\text{e}^{3\text{x}}\cos2\text{x}$

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Answer

Consider $\text{y}=\text{e}^{3\text{x}}\cos2\text{x}$
Differentiate with respect to x,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\text{e}^{3\text{x}}\cos2\text{x}$
$=\text{e}^{3\text{x}}\times\frac{\text{d}}{\text{dx}}(\cos2\text{x})+\cos2\text{x}\frac{\text{d}}{\text{dx}}(\text{e}^{3\text{x}})$
[Using chain rule]
$=\text{e}^{3\text{x}}\times(-\sin2\text{x})\frac{\text{d}}{\text{dx}}(2\text{x})+\cos2\text{xe}^{3\text{x}}\frac{\text{d}}{\text{dx}}(3\text{x})$
[Using chain rule]
$=-2\text{e}^{3\text{x}}\sin2\text{x}+3\text{e}^{3\text{x}}\cos2\text{x}$
$=\text{e}^{3\text{x}}(3\cos2\text{x}-2\sin2\text{x})$
Hence, the solution is, $\frac{\text{d}}{\text{dx}}(\text{e}^{3\text{x}}\cos2\text{x})=\text{e}^{3\text{x}}(3\cos2\text{x}-2\sin2\text{x})$

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