derivative of (-2x^2+9/(sqrt(9-x^2)))

解

\frac{d}{d x} (\frac{- 2 x ^{2} + 9}{9 - x ^{2}})

\frac{2 x ^{3} - 27 x}{( 9 - x ^{2} ) 9 - x ^{2}}

解答ステップ

\frac{d}{d x} (\frac{- 2 x ^{2} + 9}{9 - x ^{2}})

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 ^{2} + 9 ) 9 - x ^{2} - \frac{d}{d x} ( 9 - x ^{2} ) ( - 2 x ^{2} + 9 )}{( 9 - x ^{2} ) ^{2}}

\frac{d}{d x} (- 2 x^{2} + 9) = - 4 x

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

= \frac{( - 4 x ) 9 - x ^{2} - ( - \frac{x}{9 - x ^{2}} ) ( - 2 x ^{2} + 9 )}{( 9 - x ^{2} ) ^{2}}

簡素化

\frac{( - 4 x ) 9 - x ^{2} - ( - \frac{x}{9 - x ^{2}} ) ( - 2 x ^{2} + 9 )}{( 9 - x ^{2} ) ^{2}} : \frac{2 x ^{3} - 27 x}{( 9 - x ^{2} ) 9 - x ^{2}}

= \frac{2 x ^{3} - 27 x}{( 9 - x ^{2} ) 9 - x ^{2}}