derivative of (-8x^7+7x^3-2/(-4x^3))

解

\frac{d}{d x} (\frac{- 8 x ^{7} + 7 x ^{3} - 2}{- 4 x ^{3}})

- \frac{- 16 x ^{7} + 3}{2 x ^{4}}

解答ステップ

\frac{d}{d x} (\frac{- 8 x ^{7} + 7 x ^{3} - 2}{- 4 x ^{3}})

簡素化

\frac{- 8 x ^{7} + 7 x ^{3} - 2}{- 4 x ^{3}} : - \frac{- 8 x ^{7} + 7 x ^{3} - 2}{4 x ^{3}}

= \frac{d}{d x} (- \frac{- 8 x ^{7} + 7 x ^{3} - 2}{4 x ^{3}})

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= - \frac{1}{4} \frac{d}{d x} (\frac{- 8 x ^{7} + 7 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{1}{4} \cdot \frac{\frac{d}{d x} ( - 8 x ^{7} + 7 x ^{3} - 2 ) x ^{3} - \frac{d}{d x} ( x ^{3} ) ( - 8 x ^{7} + 7 x ^{3} - 2 )}{( x ^{3} ) ^{2}}

\frac{d}{d x} (- 8 x^{7} + 7 x^{3} - 2) = - 56 x^{6} + 21 x^{2}

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

= - \frac{1}{4} \cdot \frac{( - 56 x ^{6} + 21 x ^{2} ) x ^{3} - 3 x ^{2} ( - 8 x ^{7} + 7 x ^{3} - 2 )}{( x ^{3} ) ^{2}}

簡素化

- \frac{1}{4} \cdot \frac{( - 56 x ^{6} + 21 x ^{2} ) x ^{3} - 3 x ^{2} ( - 8 x ^{7} + 7 x ^{3} - 2 )}{( x ^{3} ) ^{2}} : - \frac{- 16 x ^{7} + 3}{2 x ^{4}}

= - \frac{- 16 x ^{7} + 3}{2 x ^{4}}