derivative f(x)=(ln(x))/(x^3)

解

導関数

f (x) = \frac{ln ( x )}{x ^{3}}

\frac{1 - 3 ln ( x )}{x ^{4}}

解答ステップ

\frac{d}{d x} (\frac{ln ( x )}{x ^{3}})

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 ^{3} - \frac{d}{d x} ( x ^{3} ) ln ( x )}{( x ^{3} ) ^{2}}

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

\frac{d}{d x} (x^{3}) = 3 x^{2}

= \frac{\frac{1}{x} x ^{3} - 3 x ^{2} ln ( x )}{( x ^{3} ) ^{2}}

簡素化

\frac{\frac{1}{x} x ^{3} - 3 x ^{2} ln ( x )}{( x ^{3} ) ^{2}} : \frac{1 - 3 ln ( x )}{x ^{4}}

= \frac{1 - 3 ln ( x )}{x ^{4}}