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Question
x−3 (5 + 3x)
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Solution
\[\text{ Let } u = x^{- 3} ; v = \left( 5 + 3x \right)\]
\[\text{ Then }, u = - 3 x^{- 4} ; v' = 3\]
\[\text{ Using the product rule } :\]
\[\frac{d}{dx}\left( uv \right) = uv' + vu'\]
\[\frac{d}{dx}\left[ x^3 \left( 5 + 3x \right) \right] = x^{- 3} . 3 + \left( 5 + 3x \right) \left( - 3 x^{- 4} \right)\]
\[ = 3 x^{- 3} - 15 x^{- 4} - 9 x^{- 3} \]
\[ = - 15 x^{- 4} - 6 x^{- 3}\]
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