derivative ln(x/(x^2+2))

解

導関数

ln (\frac{x}{x ^{2} + 2})

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

解答ステップ

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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