derivative of 4/(sqrt(x)+2/x+x/2)

解

\frac{d}{d x} (\frac{4}{x} + \frac{2}{x} + \frac{x}{2})

\frac{1}{2} + \frac{- 2 x ^{2} - 2 x ^{\frac{3}{2}}}{x ^{\frac{7}{2}}}

解答ステップ

\frac{d}{d x} (\frac{4}{x} + \frac{2}{x} + \frac{x}{2})

和/差の法則を適用:

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

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

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

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

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

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

分数を組み合わせる

- \frac{2}{x ^{\frac{3}{2}}} - \frac{2}{x ^{2}} : \frac{- 2 x ^{2} - 2 x ^{\frac{3}{2}}}{x ^{\frac{7}{2}}}

= \frac{1}{2} + \frac{- 2 x ^{2} - 2 x ^{\frac{3}{2}}}{x ^{\frac{7}{2}}}