(\partial)/(\partial x)((ln(x))^x)

解

\frac{\partial}{\partial x} ((ln (x))^{x})

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

解答ステップ

\frac{\partial}{\partial x} ((ln (x))^{x})

指数の規則を適用する:

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

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

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

連鎖法則を適用:

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

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

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

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

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

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