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

解

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

\frac{- 6 x ^{2} - 4 x + 4}{( 3 x ^{2} + 2 ) ^{2}}

解答ステップ

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

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

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

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

拡張

(2 - 2 x) (3 x^{2} + 2) - 6 x (2 x - x^{2}) : - 6 x^{2} - 4 x + 4

= \frac{- 6 x ^{2} - 4 x + 4}{( 3 x ^{2} + 2 ) ^{2}}