(\partial)/(\partial x)(ln(x/(1-x)))

解

\frac{\partial}{\partial x} (ln (\frac{x}{1 - x}))

\frac{1}{x ( - x + 1 )}

解答ステップ

\frac{\partial}{\partial x} (ln (\frac{x}{1 - x}))

連鎖法則を適用:

\frac{1}{\frac{x}{1 - x}} \frac{\partial}{\partial x} (\frac{x}{1 - x})

= \frac{1}{\frac{x}{1 - x}} \frac{\partial}{\partial x} (\frac{x}{1 - x})

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

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

簡素化

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

= \frac{1}{x ( - x + 1 )}