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

解答

导数

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

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

求解步骤

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

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

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

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

化简

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

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

d er i v a t i v e y = \frac{x}{( x ^{2} - 3 )}

d er i v a t i v e y = sin (x^{2} + 2)

d er i v a t i v e (2 x - 4)^{3}

d er i v a t i v e sin^{2} (x) - x

\frac{d}{d t} ((c_{1} + c, t) \cdot e^{t})