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

解

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

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

解答ステップ

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

指数の規則を適用する:

\frac{1}{a} = a^{- 1}

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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