derivative (x^3)/((x+1)^2)

解

導関数

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

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

解答ステップ

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

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

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

簡素化

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

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