derivative of (1/(x^3-2/x)(x^2+x))

解

\frac{d}{d x} ((\frac{1}{x ^{3}} - \frac{2}{x}) (x^{2} + x))

- \frac{1}{x ^{2}} - \frac{2}{x ^{3}} - 2

解答ステップ

\frac{d}{d x} ((\frac{1}{x ^{3}} - \frac{2}{x}) (x^{2} + x))

簡素化

(\frac{1}{x ^{3}} - \frac{2}{x}) (x^{2} + x) : \frac{1}{x} + \frac{1}{x ^{2}} - 2 x - 2

= \frac{d}{d x} (\frac{1}{x} + \frac{1}{x ^{2}} - 2 x - 2)

和/差の法則を適用:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d x} (\frac{1}{x}) + \frac{d}{d x} (\frac{1}{x ^{2}}) - \frac{d}{d x} (2 x) - \frac{d}{d x} (2)

\frac{d}{d x} (\frac{1}{x}) = - \frac{1}{x ^{2}}

\frac{d}{d x} (\frac{1}{x ^{2}}) = - \frac{2}{x ^{3}}

\frac{d}{d x} (2 x) = 2

\frac{d}{d x} (2) = 0

= - \frac{1}{x ^{2}} - \frac{2}{x ^{3}} - 2 - 0

簡素化

= - \frac{1}{x ^{2}} - \frac{2}{x ^{3}} - 2