derivative of (x-1/x (x^2-1/x-4))

解

\frac{d}{d x} ((x - \frac{1}{x}) (x^{2} - \frac{1}{x} - 4))

3 x^{2} - \frac{2}{x ^{3}} - \frac{4}{x ^{2}} - 5

解答ステップ

\frac{d}{d x} ((x - \frac{1}{x}) (x^{2} - \frac{1}{x} - 4))

簡素化

(x - \frac{1}{x}) (x^{2} - \frac{1}{x} - 4) : x^{3} - 1 - 5 x + \frac{1}{x ^{2}} + \frac{4}{x}

= \frac{d}{d x} (x^{3} - 1 - 5 x + \frac{1}{x ^{2}} + \frac{4}{x})

和/差の法則を適用:

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

= \frac{d}{d x} (x^{3}) - \frac{d}{d x} (1) - \frac{d}{d x} (5 x) + \frac{d}{d x} (\frac{1}{x ^{2}}) + \frac{d}{d x} (\frac{4}{x})

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

\frac{d}{d x} (1) = 0

\frac{d}{d x} (5 x) = 5

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

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

= 3 x^{2} - 0 - 5 - \frac{2}{x ^{3}} - \frac{4}{x ^{2}}

簡素化

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