derivative 1/(x+1/x)

解

derivative

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

解答ステップ

\frac{d}{d x} (\frac{1}{x + \frac{1}{x}})

指数の規則を適用する:

\frac{1}{a} = a^{- 1}

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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