derivative of (16x/((4-x^2)^2))

解

\frac{d}{d x} (\frac{16 x}{( 4 - x ^{2} ) ^{2}})

\frac{16 ( 3 x ^{2} + 4 )}{( 4 - x ^{2} ) ^{3}}

解答ステップ

\frac{d}{d x} (\frac{16 x}{( 4 - x ^{2} ) ^{2}})

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= 16 \frac{d}{d x} (\frac{x}{( 4 - x ^{2} ) ^{2}})

商の法則を適用:

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

= 16 \cdot \frac{\frac{d x}{d x} ( 4 - x ^{2} ) ^{2} - \frac{d}{d x} ( ( 4 - x ^{2} ) ^{2} ) x}{( ( 4 - x ^{2} ) ^{2} ) ^{2}}

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

\frac{d}{d x} ((4 - x^{2})^{2}) = - 4 x (4 - x^{2})

= 16 \cdot \frac{1 \cdot ( 4 - x ^{2} ) ^{2} - ( - 4 x ( 4 - x ^{2} ) ) x}{( ( 4 - x ^{2} ) ^{2} ) ^{2}}

簡素化

16 \cdot \frac{1 \cdot ( 4 - x ^{2} ) ^{2} - ( - 4 x ( 4 - x ^{2} ) ) x}{( ( 4 - x ^{2} ) ^{2} ) ^{2}} : \frac{16 ( 3 x ^{2} + 4 )}{( 4 - x ^{2} ) ^{3}}

= \frac{16 ( 3 x ^{2} + 4 )}{( 4 - x ^{2} ) ^{3}}