derivative of ln(x/2 (x-2))

解

\frac{d}{d x} (ln (\frac{x}{2}) (x - 2))

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

解答ステップ

\frac{d}{d x} (ln (\frac{x}{2}) (x - 2))

積の法則を適用:

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

f = ln (\frac{x}{2}), g = x - 2

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

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

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

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

簡素化

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

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