derivative of (x+1/x (x-1/x))

解

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

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

解答ステップ

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

積の法則を適用:

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

f = x + \frac{1}{x}, g = x - \frac{1}{x}

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

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

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

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

簡素化

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

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