derivative of (6x^2/((x^3+1)^2))

解

\frac{d}{d x} (\frac{6 x ^{2}}{( x ^{3} + 1 ) ^{2}})

\frac{12 x ( - 2 x ^{3} + 1 )}{( x ^{3} + 1 ) ^{3}}

解答ステップ

\frac{d}{d x} (\frac{6 x ^{2}}{( x ^{3} + 1 ) ^{2}})

定数を除去:

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

= 6 \frac{d}{d x} (\frac{x ^{2}}{( x ^{3} + 1 ) ^{2}})

商の法則を適用:

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

= 6 \cdot \frac{\frac{d}{d x} ( x ^{2} ) ( x ^{3} + 1 ) ^{2} - \frac{d}{d x} ( ( x ^{3} + 1 ) ^{2} ) x ^{2}}{( ( x ^{3} + 1 ) ^{2} ) ^{2}}

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

\frac{d}{d x} ((x^{3} + 1)^{2}) = 6 x^{2} (x^{3} + 1)

= 6 \cdot \frac{2 x ( x ^{3} + 1 ) ^{2} - 6 x ^{2} ( x ^{3} + 1 ) x ^{2}}{( ( x ^{3} + 1 ) ^{2} ) ^{2}}

簡素化

6 \cdot \frac{2 x ( x ^{3} + 1 ) ^{2} - 6 x ^{2} ( x ^{3} + 1 ) x ^{2}}{( ( x ^{3} + 1 ) ^{2} ) ^{2}} : \frac{12 x ( - 2 x ^{3} + 1 )}{( x ^{3} + 1 ) ^{3}}

= \frac{12 x ( - 2 x ^{3} + 1 )}{( x ^{3} + 1 ) ^{3}}