derivative of x^{4/x}

解

\frac{d}{d x} (x^{\frac{4}{x}})

4 x^{\frac{4 - 2 x}{x}} (- ln (x) + 1)

解答ステップ

\frac{d}{d x} (x^{\frac{4}{x}})

指数の規則を適用する:

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

x^{\frac{4}{x}} = e^{\frac{4}{x} l n (x)}

= \frac{d}{d x} (e^{\frac{4}{x} l n (x)})

連鎖法則を適用:

e^{\frac{4}{x} l n (x)} \frac{d}{d x} (\frac{4}{x} ln (x))

= e^{\frac{4}{x} l n (x)} \frac{d}{d x} (\frac{4}{x} ln (x))

\frac{d}{d x} (\frac{4}{x} ln (x)) = \frac{4 ( - ln ( x ) + 1 )}{x ^{2}}

= e^{\frac{4}{x} l n (x)} \frac{4 ( - ln ( x ) + 1 )}{x ^{2}}

簡素化

e^{\frac{4}{x} l n (x)} \frac{4 ( - ln ( x ) + 1 )}{x ^{2}} : 4 x^{\frac{4 - 2 x}{x}} (- ln (x) + 1)

= 4 x^{\frac{4 - 2 x}{x}} (- ln (x) + 1)