derivative f(x)=(-x^2-9)/((x^2-9)^2)

解

導関数

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

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

解答ステップ

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

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

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

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

簡素化

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

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