derivative of (x^2+6x/(10-x^2))

解

\frac{d}{d x} (\frac{x ^{2} + 6 x}{10 - x ^{2}})

\frac{6 x ^{2} + 20 x + 60}{( 10 - x ^{2} ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x ^{2} + 6 x}{10 - 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} + 6 x ) ( 10 - x ^{2} ) - \frac{d}{d x} ( 10 - x ^{2} ) ( x ^{2} + 6 x )}{( 10 - x ^{2} ) ^{2}}

\frac{d}{d x} (x^{2} + 6 x) = 2 x + 6

\frac{d}{d x} (10 - x^{2}) = - 2 x

= \frac{( 2 x + 6 ) ( 10 - x ^{2} ) - ( - 2 x ) ( x ^{2} + 6 x )}{( 10 - x ^{2} ) ^{2}}

簡素化

\frac{( 2 x + 6 ) ( 10 - x ^{2} ) - ( - 2 x ) ( x ^{2} + 6 x )}{( 10 - x ^{2} ) ^{2}} : \frac{6 x ^{2} + 20 x + 60}{( 10 - x ^{2} ) ^{2}}

= \frac{6 x ^{2} + 20 x + 60}{( 10 - x ^{2} ) ^{2}}