derivative of (9x^2/((x^2+x)^3))

解

\frac{d}{d x} (\frac{9 x ^{2}}{( x ^{2} + x ) ^{3}})

\frac{9 ( - 4 x - 1 )}{x ^{2} ( x + 1 ) ^{4}}

解答ステップ

\frac{d}{d x} (\frac{9 x ^{2}}{( x ^{2} + x ) ^{3}})

定数を除去:

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

= 9 \frac{d}{d x} (\frac{x ^{2}}{( x ^{2} + x ) ^{3}})

商の法則を適用:

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

= 9 \cdot \frac{\frac{d}{d x} ( x ^{2} ) ( x ^{2} + x ) ^{3} - \frac{d}{d x} ( ( x ^{2} + x ) ^{3} ) x ^{2}}{( ( x ^{2} + x ) ^{3} ) ^{2}}

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

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

= 9 \cdot \frac{2 x ( x ^{2} + x ) ^{3} - 3 ( x ^{2} + x ) ^{2} ( 2 x + 1 ) x ^{2}}{( ( x ^{2} + x ) ^{3} ) ^{2}}

簡素化

9 \cdot \frac{2 x ( x ^{2} + x ) ^{3} - 3 ( x ^{2} + x ) ^{2} ( 2 x + 1 ) x ^{2}}{( ( x ^{2} + x ) ^{3} ) ^{2}} : \frac{9 ( - 4 x - 1 )}{x ^{2} ( x + 1 ) ^{4}}

= \frac{9 ( - 4 x - 1 )}{x ^{2} ( x + 1 ) ^{4}}