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

Lösung

ableitung von

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

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

Schritte zur Lösung

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

Wende die Kettenregel an:

\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}})

Vereinfache

\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 )}

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