derivative of 5/x ln(x)

解

\frac{d}{d x} (\frac{5}{x} ln (x))

\frac{5 ( - ln ( x ) + 1 )}{x ^{2}}

解答ステップ

\frac{d}{d x} (\frac{5}{x} ln (x))

定数を除去:

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

= 5 \frac{d}{d x} (\frac{1}{x} ln (x))

積の法則を適用:

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

f = \frac{1}{x}, g = ln (x)

= 5 (\frac{d}{d x} (\frac{1}{x}) ln (x) + \frac{d}{d x} (ln (x)) \frac{1}{x})

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

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

= 5 ((- \frac{1}{x ^{2}}) ln (x) + \frac{1}{x} \cdot \frac{1}{x})

簡素化

5 ((- \frac{1}{x ^{2}}) ln (x) + \frac{1}{x} \cdot \frac{1}{x}) : \frac{5 ( - ln ( x ) + 1 )}{x ^{2}}

= \frac{5 ( - ln ( x ) + 1 )}{x ^{2}}