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

解

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

- \frac{14 x}{( 3 x ^{2} + 1 ) ^{2}}

解答ステップ

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

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

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

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

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

簡素化

\frac{( - 2 x ) ( 3 x ^{2} + 1 ) - 6 x ( 2 - x ^{2} )}{( 3 x ^{2} + 1 ) ^{2}} : - \frac{14 x}{( 3 x ^{2} + 1 ) ^{2}}

= - \frac{14 x}{( 3 x ^{2} + 1 ) ^{2}}