derivative of (x^{-3}+4(x^{-2}-5x))

解答

\frac{d}{d x} ((x^{- 3} + 4) (x^{- 2} - 5 x))

\frac{2 x ^{3} - 5}{x ^{6}} - 20

求解步骤

\frac{d}{d x} ((x^{- 3} + 4) (x^{- 2} - 5 x))

使用微分乘法定则:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x^{- 3} + 4, g = x^{- 2} - 5 x

= \frac{d}{d x} (x^{- 3} + 4) (x^{- 2} - 5 x) + \frac{d}{d x} (x^{- 2} - 5 x) (x^{- 3} + 4)

\frac{d}{d x} (x^{- 3} + 4) = - \frac{3}{x ^{4}}

\frac{d}{d x} (x^{- 2} - 5 x) = - \frac{2}{x ^{3}} - 5

= (- \frac{3}{x ^{4}}) (x^{- 2} - 5 x) + (- \frac{2}{x ^{3}} - 5) (x^{- 3} + 4)

化简

(- \frac{3}{x ^{4}}) (x^{- 2} - 5 x) + (- \frac{2}{x ^{3}} - 5) (x^{- 3} + 4) : \frac{2 x ^{3} - 5}{x ^{6}} - 20

= \frac{2 x ^{3} - 5}{x ^{6}} - 20

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