derivative (5-x)/((4x^2+1)^3)

解

導関数

\frac{5 - x}{( 4 x ^{2} + 1 ) ^{3}}

- \frac{- 20 x ^{2} + 120 x + 1}{( 4 x ^{2} + 1 ) ^{4}}

解答ステップ

\frac{d}{d x} (\frac{5 - x}{( 4 x ^{2} + 1 ) ^{3}})

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d x} ( 5 - x ) ( 4 x ^{2} + 1 ) ^{3} - \frac{d}{d x} ( ( 4 x ^{2} + 1 ) ^{3} ) ( 5 - x )}{( ( 4 x ^{2} + 1 ) ^{3} ) ^{2}}

\frac{d}{d x} (5 - x) = - 1

\frac{d}{d x} ((4 x^{2} + 1)^{3}) = 24 x (4 x^{2} + 1)^{2}

= \frac{( - 1 ) ( 4 x ^{2} + 1 ) ^{3} - 24 x ( 4 x ^{2} + 1 ) ^{2} ( 5 - x )}{( ( 4 x ^{2} + 1 ) ^{3} ) ^{2}}

簡素化

\frac{( - 1 ) ( 4 x ^{2} + 1 ) ^{3} - 24 x ( 4 x ^{2} + 1 ) ^{2} ( 5 - x )}{( ( 4 x ^{2} + 1 ) ^{3} ) ^{2}} : - \frac{- 20 x ^{2} + 120 x + 1}{( 4 x ^{2} + 1 ) ^{4}}

= - \frac{- 20 x ^{2} + 120 x + 1}{( 4 x ^{2} + 1 ) ^{4}}