derivative of ln(x^{1/x})

Lösung

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

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

Schritte zur Lösung

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

Wende die Kettenregel an:

\frac{1}{x ^{\frac{1}{x}}} \frac{d}{d x} (x^{\frac{1}{x}})

= \frac{1}{x ^{\frac{1}{x}}} \frac{d}{d x} (x^{\frac{1}{x}})

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

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

Vereinfache

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

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

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