derivative of e^{(1/x (ln(1-2x))})

Lösung

\frac{d}{d x} (e^{(\frac{1}{x}) (l n (1 - 2 x))})

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

Schritte zur Lösung

\frac{d}{d x} (e^{(\frac{1}{x}) (l n (1 - 2 x))})

Vereinfache

e^{(\frac{1}{x}) (l n (1 - 2 x))} : (1 - 2 x)^{\frac{1}{x}}

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

Wende Exponentenregel an:

a^{b} = e^{b l n (a)}

(1 - 2 x)^{\frac{1}{x}} = e^{\frac{1}{x} l n (1 - 2 x)}

= \frac{d}{d x} (e^{\frac{1}{x} l n (1 - 2 x)})

Wende die Kettenregel an:

e^{\frac{1}{x} l n (1 - 2 x)} \frac{d}{d x} (\frac{1}{x} ln (1 - 2 x))

= e^{\frac{1}{x} l n (1 - 2 x)} \frac{d}{d x} (\frac{1}{x} ln (1 - 2 x))

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

= e^{\frac{1}{x} l n (1 - 2 x)} (- \frac{ln ( 1 - 2 x )}{x ^{2}} - \frac{2}{x ( 1 - 2 x )})

e^{\frac{1}{x} l n (1 - 2 x)} = (1 - 2 x)^{\frac{1}{x}}

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

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