derivative of ((x^2-4^2)/((x^2+2)^3))

解答

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

\frac{2 x ( x ^{2} - 4 ) ( - x ^{2} + 16 )}{( x ^{2} + 2 ) ^{4}}

求解步骤

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

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

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

\frac{d}{d x} ((x^{2} + 2)^{3}) = 6 x (x^{2} + 2)^{2}

= \frac{4 x ( x ^{2} - 4 ) ( x ^{2} + 2 ) ^{3} - 6 x ( x ^{2} + 2 ) ^{2} ( x ^{2} - 4 ) ^{2}}{( ( x ^{2} + 2 ) ^{3} ) ^{2}}

化简

\frac{4 x ( x ^{2} - 4 ) ( x ^{2} + 2 ) ^{3} - 6 x ( x ^{2} + 2 ) ^{2} ( x ^{2} - 4 ) ^{2}}{( ( x ^{2} + 2 ) ^{3} ) ^{2}} : \frac{2 x ( x ^{2} - 4 ) ( - x ^{2} + 16 )}{( x ^{2} + 2 ) ^{4}}

= \frac{2 x ( x ^{2} - 4 ) ( - x ^{2} + 16 )}{( x ^{2} + 2 ) ^{4}}

\int \frac{1}{y - 1} d y

x \to 1 lim (1^{2} - 3)

x y^{^{'}} - 2 y = - x^{2}

\int e^{- x^{5}} d x

\int \frac{1}{t ^{2} 16 - t ^{2}} d t