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

解

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

- \frac{2 x - 1}{x ^{2} - x + 1}

解答ステップ

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

簡素化

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

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

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

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

連鎖法則を適用:

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

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

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

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

簡素化

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

= - \frac{2 x - 1}{x ^{2} - x + 1}