derivative of (5-x^2/(5+x^2))

解

\frac{d}{d x} (\frac{5 - x ^{2}}{5 + x ^{2}})

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

解答ステップ

\frac{d}{d x} (\frac{5 - x ^{2}}{5 + x ^{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} ( 5 - x ^{2} ) ( 5 + x ^{2} ) - \frac{d}{d x} ( 5 + x ^{2} ) ( 5 - x ^{2} )}{( 5 + x ^{2} ) ^{2}}

\frac{d}{d x} (5 - x^{2}) = - 2 x

\frac{d}{d x} (5 + x^{2}) = 2 x

= \frac{( - 2 x ) ( 5 + x ^{2} ) - 2 x ( 5 - x ^{2} )}{( 5 + x ^{2} ) ^{2}}

簡素化

\frac{( - 2 x ) ( 5 + x ^{2} ) - 2 x ( 5 - x ^{2} )}{( 5 + x ^{2} ) ^{2}} : - \frac{20 x}{( 5 + x ^{2} ) ^{2}}

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