derivative of (x^5/(2-x^4))

解

\frac{d}{d x} (\frac{x ^{5}}{2 - x ^{4}})

\frac{10 x ^{4} - x ^{8}}{( 2 - x ^{4} ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x ^{5}}{2 - x ^{4}})

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 ^{5} ) ( 2 - x ^{4} ) - \frac{d}{d x} ( 2 - x ^{4} ) x ^{5}}{( 2 - x ^{4} ) ^{2}}

\frac{d}{d x} (x^{5}) = 5 x^{4}

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

= \frac{5 x ^{4} ( 2 - x ^{4} ) - ( - 4 x ^{3} ) x ^{5}}{( 2 - x ^{4} ) ^{2}}

簡素化

\frac{5 x ^{4} ( 2 - x ^{4} ) - ( - 4 x ^{3} ) x ^{5}}{( 2 - x ^{4} ) ^{2}} : \frac{10 x ^{4} - x ^{8}}{( 2 - x ^{4} ) ^{2}}

= \frac{10 x ^{4} - x ^{8}}{( 2 - x ^{4} ) ^{2}}