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Find the second derivative of sin 3 x cos 5x

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y = \(\cfrac12\) [sin(5x + 3x) + sin(5x - 3x)]

y = \(\cfrac12\) sin 8x + \(\cfrac12\) sin 2x

Differentiating with respect to x

\(\cfrac{dy}{d\text x} \) = \(\cfrac82\) cos 8x + \(\cfrac22\) cos 2x

⇒ \(\cfrac{dy}{d\text x} \) = 4 cos 8x + cos 2x

Differentiating with respect to x

\(\cfrac{d^2y}{d\text x^2} \) = -32 sin 8x - 2 sin 2x

Hence Proved

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