derivative of (ln(x)/(x^4))

解

\frac{d}{d x} (\frac{ln ( x )}{x ^{4}})

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

解答ステップ

\frac{d}{d x} (\frac{ln ( x )}{x ^{4}})

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 ^{4} - \frac{d}{d x} ( x ^{4} ) ln ( x )}{( x ^{4} ) ^{2}}

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

\frac{d}{d x} (x^{4}) = 4 x^{3}

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

簡素化

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

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