derivative ((x^{1/3}-4))/(x^{7/2)}

解

導関数

\frac{( x ^{\frac{1}{3}} - 4 )}{x ^{\frac{7}{2}}}

\frac{- 19 x ^{\frac{17}{6}} + 84 x ^{\frac{5}{2}}}{6 x ^{7}}

解答ステップ

\frac{d}{d x} (\frac{x ^{\frac{1}{3}} - 4}{x ^{\frac{7}{2}}})

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d x} ( x ^{\frac{1}{3}} - 4 ) x ^{\frac{7}{2}} - \frac{d}{d x} ( x ^{\frac{7}{2}} ) ( x ^{\frac{1}{3}} - 4 )}{( x ^{\frac{7}{2}} ) ^{2}}

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

\frac{d}{d x} (x^{\frac{7}{2}}) = \frac{7 x ^{\frac{5}{2}}}{2}

= \frac{\frac{1}{3 x ^{\frac{2}{3}}} x ^{\frac{7}{2}} - \frac{7 x ^{\frac{5}{2}}}{2} ( x ^{\frac{1}{3}} - 4 )}{( x ^{\frac{7}{2}} ) ^{2}}

簡素化

\frac{\frac{1}{3 x ^{\frac{2}{3}}} x ^{\frac{7}{2}} - \frac{7 x ^{\frac{5}{2}}}{2} ( x ^{\frac{1}{3}} - 4 )}{( x ^{\frac{7}{2}} ) ^{2}} : \frac{- 19 x ^{\frac{17}{6}} + 84 x ^{\frac{5}{2}}}{6 x ^{7}}

= \frac{- 19 x ^{\frac{17}{6}} + 84 x ^{\frac{5}{2}}}{6 x ^{7}}