derivative of (4x+x^2/((2+x)^2))

解答

\frac{d}{d x} (\frac{4 x + x ^{2}}{( 2 + x ) ^{2}})

\frac{8}{( 2 + x ) ^{3}}

求解步骤

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

\frac{d}{d x} (4 x + x^{2}) = 4 + 2 x

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

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

化简

\frac{( 4 + 2 x ) ( 2 + x ) ^{2} - 2 ( 2 + x ) ( 4 x + x ^{2} )}{( ( 2 + x ) ^{2} ) ^{2}} : \frac{8}{( 2 + x ) ^{3}}

= \frac{8}{( 2 + x ) ^{3}}

\frac{d}{d x} (arctan (e^{x} x^{2}))

x \to 0 lim (\frac{4}{x})

t y^{^{'}} - 4 y = t^{6} e^{t}

d er i v a t i v e \frac{11}{( 9 x + 11 ) ^{2}}

\frac{\partial}{\partial x} (\frac{y}{( y + x ) ^{2}})