derivative of-(2x(-x^2+27)/((x^2+9)^3))

解

\frac{d}{d x} (- \frac{2 x ( - x ^{2} + 27 )}{( x ^{2} + 9 ) ^{3}})

- \frac{6 ( x ^{4} - 54 x ^{2} + 81 )}{( x ^{2} + 9 ) ^{4}}

解答ステップ

\frac{d}{d x} (- \frac{2 x ( - x ^{2} + 27 )}{( x ^{2} + 9 ) ^{3}})

定数を除去:

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

= - 2 \frac{d}{d x} (\frac{x ( - x ^{2} + 27 )}{( x ^{2} + 9 ) ^{3}})

商の法則を適用:

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

= - 2 \cdot \frac{\frac{d}{d x} ( x ( - x ^{2} + 27 ) ) ( x ^{2} + 9 ) ^{3} - \frac{d}{d x} ( ( x ^{2} + 9 ) ^{3} ) x ( - x ^{2} + 27 )}{( ( x ^{2} + 9 ) ^{3} ) ^{2}}

\frac{d}{d x} (x (- x^{2} + 27)) = - 3 x^{2} + 27

\frac{d}{d x} ((x^{2} + 9)^{3}) = 6 x (x^{2} + 9)^{2}

= - 2 \cdot \frac{( - 3 x ^{2} + 27 ) ( x ^{2} + 9 ) ^{3} - 6 x ( x ^{2} + 9 ) ^{2} x ( - x ^{2} + 27 )}{( ( x ^{2} + 9 ) ^{3} ) ^{2}}

簡素化

- 2 \cdot \frac{( - 3 x ^{2} + 27 ) ( x ^{2} + 9 ) ^{3} - 6 x ( x ^{2} + 9 ) ^{2} x ( - x ^{2} + 27 )}{( ( x ^{2} + 9 ) ^{3} ) ^{2}} : - \frac{6 ( x ^{4} - 54 x ^{2} + 81 )}{( x ^{2} + 9 ) ^{4}}

= - \frac{6 ( x ^{4} - 54 x ^{2} + 81 )}{( x ^{2} + 9 ) ^{4}}