derivative of (x-1/x ^{5/2})

解

\frac{d}{d x} ((x - \frac{1}{x})^{\frac{5}{2}})

\frac{5 ( x ^{2} + 1 ) ( x ^{2} - 1 ) ^{\frac{3}{2}}}{2 x ^{\frac{7}{2}}}

解答ステップ

\frac{d}{d x} ((x - \frac{1}{x})^{\frac{5}{2}})

連鎖法則を適用:

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

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

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

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

簡素化

\frac{5 ( x - \frac{1}{x} ) ^{\frac{3}{2}}}{2} (1 + \frac{1}{x ^{2}}) : \frac{5 ( x ^{2} + 1 ) ( x ^{2} - 1 ) ^{\frac{3}{2}}}{2 x ^{\frac{7}{2}}}

= \frac{5 ( x ^{2} + 1 ) ( x ^{2} - 1 ) ^{\frac{3}{2}}}{2 x ^{\frac{7}{2}}}