derivative of (5sqrt(x)^{ln(x)})

解答

\frac{d}{d x} ((5 x)^{l n (x)})

\frac{x ^{\frac{l n ( 25 x ) - 2}{2}} ln ( 25 x ^{2} )}{2}

求解步骤

\frac{d}{d x} ((5 x)^{l n (x)})

使用指数法则:

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

(5 x)^{l n (x)} = e^{l n (x) l n (5 x)}

= \frac{d}{d x} (e^{l n (x) l n (5 x)})

使用连锁律:

e^{l n (x) l n (5 x)} \frac{d}{d x} (ln (x) ln (5 x))

= e^{l n (x) l n (5 x)} \frac{d}{d x} (ln (x) ln (5 x))

\frac{d}{d x} (ln (x) ln (5 x)) = \frac{ln ( 5 x )}{x} + \frac{ln ( x )}{2 x}

= e^{l n (x) l n (5 x)} (\frac{ln ( 5 x )}{x} + \frac{ln ( x )}{2 x})

化简

e^{l n (x) l n (5 x)} (\frac{ln ( 5 x )}{x} + \frac{ln ( x )}{2 x}) : \frac{x ^{\frac{l n ( 25 x ) - 2}{2}} ln ( 25 x ^{2} )}{2}

= \frac{x ^{\frac{l n ( 25 x ) - 2}{2}} ln ( 25 x ^{2} )}{2}

\int sin (2 x) cos (2 x) d x

d er i v a t i v e f (x) = \frac{1}{5} x - \frac{1}{2}

\int 8 x^{7} d x

\frac{x ^{\frac{3}{2}}}{3}

in v erse l a pl a ce \frac{e ^{- 2 s}}{s ( 3 s + 2 )}