derivative of x^{(1/5 ln(x)})

解

\frac{d}{d x} (x^{(\frac{1}{5} l n (x))})

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

解答ステップ

\frac{d}{d x} (x^{(\frac{1}{5} l n (x))})

指数の規則を適用する:

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

x^{\frac{1}{5} l n (x)} = e^{\frac{1}{5} l n (x) l n (x)}

= \frac{d}{d x} (e^{\frac{1}{5} l n (x) l n (x)})

連鎖法則を適用:

e^{\frac{1}{5} l n (x) l n (x)} \frac{d}{d x} (\frac{1}{5} ln (x) ln (x))

= e^{\frac{1}{5} l n (x) l n (x)} \frac{d}{d x} (\frac{1}{5} ln (x) ln (x))

\frac{d}{d x} (\frac{1}{5} ln (x) ln (x)) = \frac{2 ln ( x )}{5 x}

= e^{\frac{1}{5} l n (x) l n (x)} \frac{2 ln ( x )}{5 x}

簡素化

e^{\frac{1}{5} l n (x) l n (x)} \frac{2 ln ( x )}{5 x} : \frac{2 e ^{\frac{l n ^{2} ( x )}{5}} ln ( x )}{5 x}

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