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

解答

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

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

求解步骤

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

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

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

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

化简

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

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

\frac{d}{d x} (e^{(x y)})

\int x + 5 d x

d er i v a t i v e \frac{- x ^{2}}{2}

\frac{d}{d x} (- 8 e^{- 4 x^{2}} x)

\int \frac{1}{1 + e ^{- y}} d y