derivative ln(sqrt((4x-4)/(5x+9)))

解

導関数

ln (\frac{4 x - 4}{5 x + 9})

\frac{7}{( x - 1 ) ( 5 x + 9 )}

解答ステップ

\frac{d}{d x} (ln (\frac{4 x - 4}{5 x + 9}))

連鎖法則を適用:

\frac{1}{\frac{4 x - 4}{5 x + 9}} \frac{d}{d x} (\frac{4 x - 4}{5 x + 9})

= \frac{1}{\frac{4 x - 4}{5 x + 9}} \frac{d}{d x} (\frac{4 x - 4}{5 x + 9})

\frac{d}{d x} (\frac{4 x - 4}{5 x + 9}) = \frac{14}{x - 1 ( 5 x + 9 ) ^{\frac{3}{2}}}

= \frac{1}{\frac{4 x - 4}{5 x + 9}} \cdot \frac{14}{x - 1 ( 5 x + 9 ) ^{\frac{3}{2}}}

簡素化

\frac{1}{\frac{4 x - 4}{5 x + 9}} \cdot \frac{14}{x - 1 ( 5 x + 9 ) ^{\frac{3}{2}}} : \frac{7}{( x - 1 ) ( 5 x + 9 )}

= \frac{7}{( x - 1 ) ( 5 x + 9 )}