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

解

\frac{d}{d x} ((ln (x))^{s e c (x)})

ln^{s e c (x)} (x) (sec (x) tan (x) ln (ln (x)) + \frac{sec ( x )}{x ln ( x )})

解答ステップ

\frac{d}{d x} ((ln (x))^{s e c (x)})

指数の規則を適用する:

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

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

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

連鎖法則を適用:

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

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

\frac{d}{d x} (sec (x) ln (ln (x))) = sec (x) tan (x) ln (ln (x)) + \frac{sec ( x )}{x ln ( x )}

= e^{s e c (x) l n (l n (x))} (sec (x) tan (x) ln (ln (x)) + \frac{sec ( x )}{x ln ( x )})

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

= ln^{s e c (x)} (x) (sec (x) tan (x) ln (ln (x)) + \frac{sec ( x )}{x ln ( x )})