derivative y=x^{1/x}

解

導関数

y = x^{\frac{1}{x}}

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

解答ステップ

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

指数の規則を適用する:

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

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

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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