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

解

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

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

解答ステップ

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

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

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

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

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

簡素化

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

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