derivative ln(1+3/x+5/(x^2))

解

導関数

ln (1 + \frac{3}{x} + \frac{5}{x ^{2}})

\frac{- 3 x - 10}{x ( x ^{2} + 3 x + 5 )}

解答ステップ

\frac{d}{d x} (ln (1 + \frac{3}{x} + \frac{5}{x ^{2}}))

連鎖法則を適用:

\frac{1}{1 + \frac{3}{x} + \frac{5}{x ^{2}}} \frac{d}{d x} (1 + \frac{3}{x} + \frac{5}{x ^{2}})

= \frac{1}{1 + \frac{3}{x} + \frac{5}{x ^{2}}} \frac{d}{d x} (1 + \frac{3}{x} + \frac{5}{x ^{2}})

\frac{d}{d x} (1 + \frac{3}{x} + \frac{5}{x ^{2}}) = - \frac{3}{x ^{2}} - \frac{10}{x ^{3}}

= \frac{1}{1 + \frac{3}{x} + \frac{5}{x ^{2}}} (- \frac{3}{x ^{2}} - \frac{10}{x ^{3}})

簡素化

\frac{1}{1 + \frac{3}{x} + \frac{5}{x ^{2}}} (- \frac{3}{x ^{2}} - \frac{10}{x ^{3}}) : \frac{- 3 x - 10}{x ( x ^{2} + 3 x + 5 )}

= \frac{- 3 x - 10}{x ( x ^{2} + 3 x + 5 )}