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

解

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

\frac{- 8 x ^{3} - 15}{x ^{6}}

解答ステップ

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

積の法則を適用:

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

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

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

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

\frac{d}{d x} (4 + 3 x^{- 3}) = - \frac{9}{x ^{4}}

= (- \frac{2}{x ^{3}}) (4 + 3 x^{- 3}) + (- \frac{9}{x ^{4}}) x^{- 2}

簡素化

(- \frac{2}{x ^{3}}) (4 + 3 x^{- 3}) + (- \frac{9}{x ^{4}}) x^{- 2} : \frac{- 8 x ^{3} - 15}{x ^{6}}

= \frac{- 8 x ^{3} - 15}{x ^{6}}