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Question
x4 (5 sin x − 3 cos x)
Solution
\[\text{ Let } u = x^4 ; v = 5 \sin x - 3 \cos x\]
\[\text{ Then }, u' = 4 x^3 ; v' = 5 \cos x - 3 ( - \sin x) = 5 \cos x + 3 \sin x \]
\[\text{ According to the product rule }:\]
\[\frac{d}{dx}\left( uv \right) = u v' + v u'\]
\[\frac{d}{dx}\left( x^4 \left( 5 \sin x - 3 \cos x \right) \right) = x^4 \left( 5 \cos x + 3 \sin x \right) + \left( 5 \sin x - 3 \cos x \right) 4 x^3 \]
\[ = x^3 \left( 5x \cos x + 3 x \sin x + 20 \sin x - 12 \cos x \right)\]
\[ = x^3 \left( \left( 3x + 20 \right) \sin x + \left( 5x - 12 \right) \cos x \right)\]
\[ = 3 x^4 \sin x + 20 x^3 \sin x + 5x \cos x - 12 \cos x\]
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