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

解答

导数

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

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

求解步骤

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

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

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

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

化简

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

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

\frac{\partial}{\partial x} (4 e^{yz} cos (x yz))

x \to 7 lim (\frac{- 2}{7 - x})

t an g e n t y = \frac{6 x}{x + 2}, (4, 4)

\int \frac{( x - 2 )}{x ^{2} - 7 x + 12} d x

\int \frac{1}{x lo g _{10} ( x )} d x