Find the second derivative of sin 3x cos 5x.

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Solution

Let $y = sin 3 x cos 5 x$ …….. $(1)$
we have to find $\frac{d^{2} y}{d x^{2}} = ?$
on differentiating equation $(1)$ wrt x we get
$\frac{d y}{d x} \frac{d}{d x} (sin 3 x cos 5 x)$
$\frac{d y}{d x} = 3 cos 3 x cos 5 x - 5 sin 5 x sin 3 x$ [using multiplication rule]
again differentiating wrt x we get
$\frac{d^{2} y}{d x^{2}} = 3 \frac{d}{d x} (cos 3 x cos 5 x) - 5 \frac{d}{d x} (sin 5 x sin 3 x)$
$= 3 [- 3 sin 3 x cos 5 x - 5 cos 3 x sin 5 x] - 5 [5 cos 5 x sin 3 x + 3 cos 3 x sin 5 x]$
$\Rightarrow - 9 sin 3 x cos 5 x + 15 cos 3 x sin 5 x - 25 cos 5 x sin 3 x - 15 cos 3 x sin 5 x$
$\Rightarrow - 34 sin 3 x cos 5 x$ .

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