derivative of (1-2x^{1/x})

解

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

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

解答ステップ

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

指数の規則を適用する:

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

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

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

連鎖法則を適用:

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

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

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

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

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

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