derivative of ln(\sqrt[3]{(3x/(x+2)}))

解

\frac{d}{d x} (ln (3 \frac{3 x}{x + 2}))

\frac{2}{3 3 \cdot 3 ^{\frac{2}{3}} 3 x x ^{\frac{2}{3}} ( x + 2 )}

解答ステップ

\frac{d}{d x} (ln (3 \frac{3 x}{x + 2}))

連鎖法則を適用:

\frac{1}{3 \frac{3 x}{x + 2}} \frac{d}{d x} (3 \frac{3 x}{x + 2})

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

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

= \frac{1}{3 \frac{3 x}{x + 2}} \cdot \frac{2}{3 ^{\frac{2}{3}} x ^{\frac{2}{3}} ( x + 2 ) ^{\frac{4}{3}}}

簡素化

\frac{1}{3 \frac{3 x}{x + 2}} \cdot \frac{2}{3 ^{\frac{2}{3}} x ^{\frac{2}{3}} ( x + 2 ) ^{\frac{4}{3}}} : \frac{2}{3 3 \cdot 3 ^{\frac{2}{3}} 3 x x ^{\frac{2}{3}} ( x + 2 )}

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