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Question
sin x cos x
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Solution
\[\text{ Let } u = \sin x; v = \cos x\]
\[\text{ Then }, u' = \cos x; v' = - \sin x\]
\[\text{ Using theproduct rule }:\]
\[\frac{d}{dx}\left( uv \right) = uv' + vu'\]
\[\frac{d}{dx}\left( \sin x \cos x \right) = \sin x \left( - \sin x \right) + \cos x . \cos x\]
\[ = - \sin^2 x + \cos^2 x\]
\[ = \cos 2x\]
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