derivative of xln(1-x)

解

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

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

解答ステップ

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

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = ln (1 - x)

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

\frac{d x}{d x} = 1

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

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

簡素化

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

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