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

解

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

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

解答ステップ

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

指数の規則を適用する:

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

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

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

連鎖法則を適用:

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

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

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

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

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

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