derivative y=(7x^3-3x^2-9)/(x^8)

解

導関数

y = \frac{7 x ^{3} - 3 x ^{2} - 9}{x ^{8}}

\frac{- 35 x ^{3} + 18 x ^{2} + 72}{x ^{9}}

解答ステップ

\frac{d}{d x} (\frac{7 x ^{3} - 3 x ^{2} - 9}{x ^{8}})

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d x} ( 7 x ^{3} - 3 x ^{2} - 9 ) x ^{8} - \frac{d}{d x} ( x ^{8} ) ( 7 x ^{3} - 3 x ^{2} - 9 )}{( x ^{8} ) ^{2}}

\frac{d}{d x} (7 x^{3} - 3 x^{2} - 9) = 21 x^{2} - 6 x

\frac{d}{d x} (x^{8}) = 8 x^{7}

= \frac{( 21 x ^{2} - 6 x ) x ^{8} - 8 x ^{7} ( 7 x ^{3} - 3 x ^{2} - 9 )}{( x ^{8} ) ^{2}}

簡素化

\frac{( 21 x ^{2} - 6 x ) x ^{8} - 8 x ^{7} ( 7 x ^{3} - 3 x ^{2} - 9 )}{( x ^{8} ) ^{2}} : \frac{- 35 x ^{3} + 18 x ^{2} + 72}{x ^{9}}

= \frac{- 35 x ^{3} + 18 x ^{2} + 72}{x ^{9}}