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

解

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

\frac{12 x ^{2}}{( 2 - x ^{3} ) ^{2}}

解答ステップ

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

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

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

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

簡素化

\frac{3 x ^{2} ( 2 - x ^{3} ) - ( - 3 x ^{2} ) ( 2 + x ^{3} )}{( 2 - x ^{3} ) ^{2}} : \frac{12 x ^{2}}{( 2 - x ^{3} ) ^{2}}

= \frac{12 x ^{2}}{( 2 - x ^{3} ) ^{2}}