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

解

導関数

\frac{x}{9 - x ^{2}}

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

解答ステップ

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

\frac{d x}{d x} = 1

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

= \frac{1 \cdot 9 - x ^{2} - ( - \frac{x}{9 - x ^{2}} ) x}{( 9 - x ^{2} ) ^{2}}

簡素化

\frac{1 \cdot 9 - x ^{2} - ( - \frac{x}{9 - x ^{2}} ) x}{( 9 - x ^{2} ) ^{2}} : \frac{9}{( 9 - x ^{2} ) 9 - x ^{2}}

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