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

解答

导数

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

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

求解步骤

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

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

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

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

乘开

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

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

t an g e n t \frac{3 x}{4 - x ^{2}} = y, at x = 1

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