derivative of (ln(x)/(5x^3))

解

\frac{d}{d x} (\frac{ln ( x )}{5 x ^{3}})

\frac{1 - 3 ln ( x )}{5 x ^{4}}

解答ステップ

\frac{d}{d x} (\frac{ln ( x )}{5 x ^{3}})

定数を除去:

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

= \frac{1}{5} \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{1}{5} \cdot \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{1}{5} \cdot \frac{\frac{1}{x} x ^{3} - 3 x ^{2} ln ( x )}{( x ^{3} ) ^{2}}

簡素化

\frac{1}{5} \cdot \frac{\frac{1}{x} x ^{3} - 3 x ^{2} ln ( x )}{( x ^{3} ) ^{2}} : \frac{1 - 3 ln ( x )}{5 x ^{4}}

= \frac{1 - 3 ln ( x )}{5 x ^{4}}