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

解

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

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

解答ステップ

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

指数の規則を適用する:

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

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

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

連鎖法則を適用:

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

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

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

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

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

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