derivative 2^{1/x}

Lösung

ableitung von

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

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

Schritte zur Lösung

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

Wende Exponentenregel an:

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

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

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

Wende die Kettenregel an:

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

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

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

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

Vereinfache

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

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

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