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

解

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

x^{l n (l n (x)) - 1} (ln (ln (x)) + 1)

解答ステップ

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

指数の規則を適用する:

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

(ln (x))^{l n (x)} = e^{l n (x) l n (l n (x))}

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

連鎖法則を適用:

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

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

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

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

簡素化

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

= x^{l n (l n (x)) - 1} (ln (ln (x)) + 1)