derivative of-(ln(1+6x)/x)

解

\frac{d}{d x} (- \frac{ln ( 1 + 6 x )}{x})

- \frac{6 x - ln ( 1 + 6 x ) ( 1 + 6 x )}{x ^{2} ( 1 + 6 x )}

解答ステップ

\frac{d}{d x} (- \frac{ln ( 1 + 6 x )}{x})

定数を除去:

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

= - \frac{d}{d x} (\frac{ln ( 1 + 6 x )}{x})

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= - \frac{\frac{d}{d x} ( ln ( 1 + 6 x ) ) x - \frac{d x}{d x} ln ( 1 + 6 x )}{x ^{2}}

\frac{d}{d x} (ln (1 + 6 x)) = \frac{6}{1 + 6 x}

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

= - \frac{\frac{6}{1 + 6 x} x - 1 \cdot ln ( 1 + 6 x )}{x ^{2}}

簡素化

- \frac{\frac{6}{1 + 6 x} x - 1 \cdot ln ( 1 + 6 x )}{x ^{2}} : - \frac{6 x - ln ( 1 + 6 x ) ( 1 + 6 x )}{x ^{2} ( 1 + 6 x )}

= - \frac{6 x - ln ( 1 + 6 x ) ( 1 + 6 x )}{x ^{2} ( 1 + 6 x )}