derivative of (\sqrt[3]{x}/({f)(x)})

解

\frac{d}{d x} (\frac{3 x}{f ( x )})

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

解答ステップ

\frac{d}{d x} (\frac{3 x}{f ( x )})

商の法則を適用:

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

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

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

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

簡素化

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

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