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

解

導関数

(x^{- 2} + x^{- 3}) (x^{5} - 2 x^{2})

3 x^{2} + 2 x + \frac{2}{x ^{2}}

解答ステップ

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

積の法則を適用:

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

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

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

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

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

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

簡素化

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

= 3 x^{2} + 2 x + \frac{2}{x ^{2}}