derivative f(x)=(1-x^2)/((1+x^2)^2)

解

導関数

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

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

解答ステップ

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

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

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

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

簡素化

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

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