derivative (1-2x^2)/(x^4-2x^2+6)

解

導関数

\frac{1 - 2 x ^{2}}{x ^{4} - 2 x ^{2} + 6}

\frac{4 x ^{5} - 4 x ^{3} - 20 x}{( x ^{4} - 2 x ^{2} + 6 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{1 - 2 x ^{2}}{x ^{4} - 2 x ^{2} + 6})

商の法則を適用:

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

= \frac{\frac{d}{d x} ( 1 - 2 x ^{2} ) ( x ^{4} - 2 x ^{2} + 6 ) - \frac{d}{d x} ( x ^{4} - 2 x ^{2} + 6 ) ( 1 - 2 x ^{2} )}{( x ^{4} - 2 x ^{2} + 6 ) ^{2}}

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

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

= \frac{( - 4 x ) ( x ^{4} - 2 x ^{2} + 6 ) - ( 4 x ^{3} - 4 x ) ( 1 - 2 x ^{2} )}{( x ^{4} - 2 x ^{2} + 6 ) ^{2}}

簡素化

\frac{( - 4 x ) ( x ^{4} - 2 x ^{2} + 6 ) - ( 4 x ^{3} - 4 x ) ( 1 - 2 x ^{2} )}{( x ^{4} - 2 x ^{2} + 6 ) ^{2}} : \frac{4 x ^{5} - 4 x ^{3} - 20 x}{( x ^{4} - 2 x ^{2} + 6 ) ^{2}}

= \frac{4 x ^{5} - 4 x ^{3} - 20 x}{( x ^{4} - 2 x ^{2} + 6 ) ^{2}}