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

解

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

\frac{18 x}{( 9 - x ^{2} ) ^{2}}

解答ステップ

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

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

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

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

簡素化

\frac{2 x ( 9 - x ^{2} ) - ( - 2 x ) x ^{2}}{( 9 - x ^{2} ) ^{2}} : \frac{18 x}{( 9 - x ^{2} ) ^{2}}

= \frac{18 x}{( 9 - x ^{2} ) ^{2}}