derivative of (1+x/(x^3))

解

\frac{d}{d x} (\frac{1 + x}{x ^{3}})

\frac{- 2 x - 3}{x ^{4}}

解答ステップ

\frac{d}{d x} (\frac{1 + 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} ( 1 + x ) x ^{3} - \frac{d}{d x} ( x ^{3} ) ( 1 + x )}{( x ^{3} ) ^{2}}

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

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

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

簡素化

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

= \frac{- 2 x - 3}{x ^{4}}