Find the derivative of the following:
$x^{5} (3 - 6 x^{- 9})$ .

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Solution

Let $f (x) = x^{5} (3 - 6 x^{- 9})$
By Leibnitz product rule
$f^{^{'}} (x) = x^{5} \frac{d}{d x} (3 - 6 x^{- 9}) + (3 - 6 x^{- 9}) \frac{d}{d x} (x^{5})$
$= x^{3} {0 - 6 (- 9) x^{- 9 - 1}} + (3 - 6 x^{- 9}) (5 x^{4})$
$= x^{3} (54 x^{- 10}) + 15 x^{4} - 30 x^{- 5}$
$= 54 x^{- 7} + 15 x^{4} - 30 x^{- 5}$

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