derivative x^{x/3}

解

導関数

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

\frac{x ^{\frac{x}{3}} ( ln ( x ) + 1 )}{3}

解答ステップ

\frac{d}{d x} (x^{\frac{x}{3}})

指数の規則を適用する:

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

x^{\frac{x}{3}} = e^{\frac{x}{3} l n (x)}

= \frac{d}{d x} (e^{\frac{x}{3} l n (x)})

連鎖法則を適用:

e^{\frac{x}{3} l n (x)} \frac{d}{d x} (\frac{x}{3} ln (x))

= e^{\frac{x}{3} l n (x)} \frac{d}{d x} (\frac{x}{3} ln (x))

\frac{d}{d x} (\frac{x}{3} ln (x)) = \frac{ln ( x ) + 1}{3}

= e^{\frac{x}{3} l n (x)} \frac{ln ( x ) + 1}{3}

簡素化

e^{\frac{x}{3} l n (x)} \frac{ln ( x ) + 1}{3} : \frac{x ^{\frac{x}{3}} ( ln ( x ) + 1 )}{3}

= \frac{x ^{\frac{x}{3}} ( ln ( x ) + 1 )}{3}