derivative (ln(x))/(x^5)

解

導関数

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

\frac{1 - 5 ln ( x )}{x ^{6}}

解答ステップ

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

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

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

\frac{d}{d x} (x^{5}) = 5 x^{4}

= \frac{\frac{1}{x} x ^{5} - 5 x ^{4} ln ( x )}{( x ^{5} ) ^{2}}

簡素化

\frac{\frac{1}{x} x ^{5} - 5 x ^{4} ln ( x )}{( x ^{5} ) ^{2}} : \frac{1 - 5 ln ( x )}{x ^{6}}

= \frac{1 - 5 ln ( x )}{x ^{6}}