derivative a^{1/x}

解

導関数

a^{\frac{1}{x}}

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

解答ステップ

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

指数の規則を適用する:

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

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

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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