derivative of ln((x^2+4/(x^2+1)))

解

\frac{d}{d x} (ln (\frac{x ^{2} + 4}{x ^{2} + 1}))

- \frac{6 x}{( x ^{2} + 4 ) ( x ^{2} + 1 )}

解答ステップ

\frac{d}{d x} (ln (\frac{x ^{2} + 4}{x ^{2} + 1}))

連鎖法則を適用:

\frac{1}{\frac{x ^{2} + 4}{x ^{2} + 1}} \frac{d}{d x} (\frac{x ^{2} + 4}{x ^{2} + 1})

= \frac{1}{\frac{x ^{2} + 4}{x ^{2} + 1}} \frac{d}{d x} (\frac{x ^{2} + 4}{x ^{2} + 1})

\frac{d}{d x} (\frac{x ^{2} + 4}{x ^{2} + 1}) = - \frac{6 x}{( x ^{2} + 1 ) ^{2}}

= \frac{1}{\frac{x ^{2} + 4}{x ^{2} + 1}} (- \frac{6 x}{( x ^{2} + 1 ) ^{2}})

簡素化

\frac{1}{\frac{x ^{2} + 4}{x ^{2} + 1}} (- \frac{6 x}{( x ^{2} + 1 ) ^{2}}) : - \frac{6 x}{( x ^{2} + 4 ) ( x ^{2} + 1 )}

= - \frac{6 x}{( x ^{2} + 4 ) ( x ^{2} + 1 )}