derivative of ln((x^3+1/(x^2+1)))

解

\frac{d}{d x} (ln (\frac{x ^{3} + 1}{x ^{2} + 1}))

\frac{x ^{4} + 3 x ^{2} - 2 x}{( x ^{3} + 1 ) ( x ^{2} + 1 )}

解答ステップ

\frac{d}{d x} (ln (\frac{x ^{3} + 1}{x ^{2} + 1}))

連鎖法則を適用:

\frac{1}{\frac{x ^{3} + 1}{x ^{2} + 1}} \frac{d}{d x} (\frac{x ^{3} + 1}{x ^{2} + 1})

= \frac{1}{\frac{x ^{3} + 1}{x ^{2} + 1}} \frac{d}{d x} (\frac{x ^{3} + 1}{x ^{2} + 1})

\frac{d}{d x} (\frac{x ^{3} + 1}{x ^{2} + 1}) = \frac{x ^{4} + 3 x ^{2} - 2 x}{( x ^{2} + 1 ) ^{2}}

= \frac{1}{\frac{x ^{3} + 1}{x ^{2} + 1}} \cdot \frac{x ^{4} + 3 x ^{2} - 2 x}{( x ^{2} + 1 ) ^{2}}

簡素化

\frac{1}{\frac{x ^{3} + 1}{x ^{2} + 1}} \cdot \frac{x ^{4} + 3 x ^{2} - 2 x}{( x ^{2} + 1 ) ^{2}} : \frac{x ^{4} + 3 x ^{2} - 2 x}{( x ^{3} + 1 ) ( x ^{2} + 1 )}

= \frac{x ^{4} + 3 x ^{2} - 2 x}{( x ^{3} + 1 ) ( x ^{2} + 1 )}