derivative of ln(x+y-ln(x-y))

解

\frac{d}{d x} (ln (x + y) - ln (x - y))

\frac{2 x \frac{d y}{d x} - 2 y}{( x + y ) ( x - y )}

解答ステップ

\frac{d}{d x} (ln (x + y) - ln (x - y))

和/差の法則を適用:

(f \pm g)^{'} = f^{'} \pm g^{'}

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

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

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

= \frac{1 + \frac{d y}{d x}}{x + y} - \frac{1 - \frac{d y}{d x}}{x - y}

簡素化

\frac{1 + \frac{d y}{d x}}{x + y} - \frac{1 - \frac{d y}{d x}}{x - y} : \frac{2 x \frac{d y}{d x} - 2 y}{( x + y ) ( x - y )}

= \frac{2 x \frac{d y}{d x} - 2 y}{( x + y ) ( x - y )}