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

解答

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

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

求解步骤

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

使用微分除法定则:

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

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

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

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

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

化简

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

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