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

解答

\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 )}

求解步骤

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

使用指数法则:

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

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

使用连锁律:

- \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))

化简

- \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 )}

\int 8 sin^{2} (t + \frac{π}{12}) d t

x y^{^{'}} = y + 6 x^{2} sin (x), y (π) = 0

x \to - 1 lim (x^{2} - 3 x - 4)

x \to 4 lim (\frac{3}{( x - 4 ) ^{2}})

\int cos^{4} (\frac{1}{2} x) d x