derivative f(x)=ln(x+ln(x))

解

導関数

f (x) = ln (x + ln (x))

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

解答ステップ

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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