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

解

\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}}

解答ステップ

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

商の法則を適用:

(\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}}

簡素化

\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}}