Differentiate

$e^{3 x} \cos 2 x$

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Solution

$Let y = e^{3 x} \cos 2 x Differentiate it with respect to x we get, \frac{d y}{d x} = \frac{d}{d x} (e^{3 x} \cos 2 x) = e^{3 x} \times \frac{d}{d x} (\cos 2 x) + \cos 2 x \frac{d}{d x} (e^{3 x}) [Using product rule] = e^{3 x} \times (- \sin 2 x) \frac{d}{d x} (2 x) + \cos 2 x e^{3 x} \frac{d}{d x} (3 x) [Using chain rule] = - 2 e^{3 x} \sin 2 x + 3 e^{3 x} \cos 2 x = e^{3 x} (3 \cos 2 x - 2 \sin 2 x) So, \frac{d}{d x} (e^{3 x} \cos 2 x) = e^{3 x} (3 \cos 2 x - 2 \sin 2 x)$

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