Find the second order derivatives of $e^{x} sin 5 x$

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Solution

Let $y = e^{x} sin 5 x$
$\Rightarrow \frac{d y}{d x} = \frac{d}{d x} (e^{x} sin 5 x) = sin 5 x . \frac{d}{d x} (e^{x}) + e^{x} \frac{d}{d x} (s i n 5 x)$
$= e^{x} (sin 5 x + 5 cos 5 x)$
$∴ \frac{d^{2} y}{d x^{2}} = \frac{d}{d x} [e^{x} (sin 5 x + 5 cos 5 x)]$
$= (sin 5 x + 5 cos 5 x) \frac{d}{d x} (e^{x}) + e^{x} . \frac{d}{d x} (sin 5 x + 5 cos 5 x)$
$= (sin 5 x + 5 cos 5 x) e^{x} + e^{x} [cos 5 x \frac{d}{d x} (5 x) + 5 (- sin 5 x) . \frac{d}{d x} (5 x)]$
$= e^{x} (sin 5 x + 5 cos 5 x) + e^{x} (5 cos 5 x - 25 sin 5 x)$
$= e^{x} (10 cos 5 x - 24 sin 5 x) = 2 e^{x} (5 cos 5 x - 12 sin 5 x)$

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