derivative of (x^2+1/(sqrt(3-x^2)))

解

\frac{d}{d x} (\frac{x ^{2} + 1}{3 - x ^{2}})

\frac{- x ^{3} + 7 x}{( 3 - x ^{2} ) 3 - x ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x ^{2} + 1}{3 - 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} + 1 ) 3 - x ^{2} - \frac{d}{d x} ( 3 - x ^{2} ) ( x ^{2} + 1 )}{( 3 - x ^{2} ) ^{2}}

\frac{d}{d x} (x^{2} + 1) = 2 x

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

= \frac{2 x 3 - x ^{2} - ( - \frac{x}{3 - x ^{2}} ) ( x ^{2} + 1 )}{( 3 - x ^{2} ) ^{2}}

簡素化

\frac{2 x 3 - x ^{2} - ( - \frac{x}{3 - x ^{2}} ) ( x ^{2} + 1 )}{( 3 - x ^{2} ) ^{2}} : \frac{- x ^{3} + 7 x}{( 3 - x ^{2} ) 3 - x ^{2}}

= \frac{- x ^{3} + 7 x}{( 3 - x ^{2} ) 3 - x ^{2}}