derivative of ln(\sqrt[6]{(x-5/(x+5)}))

解

\frac{d}{d x} (ln (6 \frac{x - 5}{x + 5}))

\frac{5}{3 ( x + 5 ) 6 x - 5 ( x - 5 ) ^{\frac{5}{6}}}

解答ステップ

\frac{d}{d x} (ln (6 \frac{x - 5}{x + 5}))

連鎖法則を適用:

\frac{1}{6 \frac{x - 5}{x + 5}} \frac{d}{d x} (6 \frac{x - 5}{x + 5})

= \frac{1}{6 \frac{x - 5}{x + 5}} \frac{d}{d x} (6 \frac{x - 5}{x + 5})

\frac{d}{d x} (6 \frac{x - 5}{x + 5}) = \frac{5}{3 ( x - 5 ) ^{\frac{5}{6}} ( x + 5 ) ^{\frac{7}{6}}}

= \frac{1}{6 \frac{x - 5}{x + 5}} \cdot \frac{5}{3 ( x - 5 ) ^{\frac{5}{6}} ( x + 5 ) ^{\frac{7}{6}}}

簡素化

\frac{1}{6 \frac{x - 5}{x + 5}} \cdot \frac{5}{3 ( x - 5 ) ^{\frac{5}{6}} ( x + 5 ) ^{\frac{7}{6}}} : \frac{5}{3 ( x + 5 ) 6 x - 5 ( x - 5 ) ^{\frac{5}{6}}}

= \frac{5}{3 ( x + 5 ) 6 x - 5 ( x - 5 ) ^{\frac{5}{6}}}