derivative of ln((9x/(x^2+15)))

解

\frac{d}{d x} (ln (\frac{9 x}{x ^{2} + 15}))

\frac{- x ^{2} + 15}{x ( x ^{2} + 15 )}

解答ステップ

\frac{d}{d x} (ln (\frac{9 x}{x ^{2} + 15}))

連鎖法則を適用:

\frac{1}{\frac{9 x}{x ^{2} + 15}} \frac{d}{d x} (\frac{9 x}{x ^{2} + 15})

= \frac{1}{\frac{9 x}{x ^{2} + 15}} \frac{d}{d x} (\frac{9 x}{x ^{2} + 15})

\frac{d}{d x} (\frac{9 x}{x ^{2} + 15}) = \frac{9 ( - x ^{2} + 15 )}{( x ^{2} + 15 ) ^{2}}

= \frac{1}{\frac{9 x}{x ^{2} + 15}} \cdot \frac{9 ( - x ^{2} + 15 )}{( x ^{2} + 15 ) ^{2}}

簡素化

\frac{1}{\frac{9 x}{x ^{2} + 15}} \cdot \frac{9 ( - x ^{2} + 15 )}{( x ^{2} + 15 ) ^{2}} : \frac{- x ^{2} + 15}{x ( x ^{2} + 15 )}

= \frac{- x ^{2} + 15}{x ( x ^{2} + 15 )}