derivative of ln(x/(x^2+5))

解

\frac{d}{d x} (ln (\frac{x}{x ^{2} + 5}))

\frac{- x ^{2} + 5}{x ( x ^{2} + 5 )}

解答ステップ

\frac{d}{d x} (ln (\frac{x}{x ^{2} + 5}))

連鎖法則を適用:

\frac{1}{\frac{x}{x ^{2} + 5}} \frac{d}{d x} (\frac{x}{x ^{2} + 5})

= \frac{1}{\frac{x}{x ^{2} + 5}} \frac{d}{d x} (\frac{x}{x ^{2} + 5})

\frac{d}{d x} (\frac{x}{x ^{2} + 5}) = \frac{- x ^{2} + 5}{( x ^{2} + 5 ) ^{2}}

= \frac{1}{\frac{x}{x ^{2} + 5}} \cdot \frac{- x ^{2} + 5}{( x ^{2} + 5 ) ^{2}}

簡素化

\frac{1}{\frac{x}{x ^{2} + 5}} \cdot \frac{- x ^{2} + 5}{( x ^{2} + 5 ) ^{2}} : \frac{- x ^{2} + 5}{x ( x ^{2} + 5 )}

= \frac{- x ^{2} + 5}{x ( x ^{2} + 5 )}