Find the derivative of the following:
$x^{- 4} (3 - 4 x^{- 5})$ .

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Solution

Let $f (x) = x^{- 4} (3 - 4 x^{- 5})$
By Leibnitz product rule
$f^{^{'}} (x) = x^{- 4} \frac{d}{d x} (3 - 4 x^{- 5}) + (3 - 4 x^{- 5}) \frac{d}{d x} (x^{- 4})$
$= x^{- 4} {0 - 4 (- 5) x^{- 5 - 3}} + (3 - 4 x^{- 5}) (- 4) x^{- 4 - 1}$
$= x^{- 4} (20 x^{- 6}) + (3 - 4 x^{- 5}) (- 4 x^{- 5})$
$= 20 x^{- 10} - 12 x^{- 5} + 16 x^{- 10}$
$= 36 x^{- 10} - 12 x^{- 5}$

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