derivative of (ln(x)dx)

解

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

d (1 + ln (x))

解答ステップ

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

d

を定数として扱う

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

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

積の法則を適用:

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

f = ln (x), g = x

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

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

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

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

簡素化

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

= d (1 + ln (x))