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

解

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

\frac{100 x}{( x ^{2} - 25 ) ( x ^{2} + 25 )}

解答ステップ

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

簡素化

ln (\frac{x ^{2} - 5 ^{2}}{x ^{2} + 5 ^{2}}) : ln (\frac{x ^{2} - 25}{x ^{2} + 25})

= \frac{d}{d x} (ln (\frac{x ^{2} - 25}{x ^{2} + 25}))

連鎖法則を適用:

\frac{1}{\frac{x ^{2} - 25}{x ^{2} + 25}} \frac{d}{d x} (\frac{x ^{2} - 25}{x ^{2} + 25})

= \frac{1}{\frac{x ^{2} - 25}{x ^{2} + 25}} \frac{d}{d x} (\frac{x ^{2} - 25}{x ^{2} + 25})

\frac{d}{d x} (\frac{x ^{2} - 25}{x ^{2} + 25}) = \frac{100 x}{( x ^{2} + 25 ) ^{2}}

= \frac{1}{\frac{x ^{2} - 25}{x ^{2} + 25}} \cdot \frac{100 x}{( x ^{2} + 25 ) ^{2}}

簡素化

\frac{1}{\frac{x ^{2} - 25}{x ^{2} + 25}} \cdot \frac{100 x}{( x ^{2} + 25 ) ^{2}} : \frac{100 x}{( x ^{2} - 25 ) ( x ^{2} + 25 )}

= \frac{100 x}{( x ^{2} - 25 ) ( x ^{2} + 25 )}