derivative of (2x+4x^2/((1+4x)^2))

解

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

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

解答ステップ

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

\frac{d}{d x} (2 x + 4 x^{2}) = 2 + 8 x

\frac{d}{d x} ((1 + 4 x)^{2}) = 8 (1 + 4 x)

= \frac{( 2 + 8 x ) ( 1 + 4 x ) ^{2} - 8 ( 1 + 4 x ) ( 2 x + 4 x ^{2} )}{( ( 1 + 4 x ) ^{2} ) ^{2}}

簡素化

\frac{( 2 + 8 x ) ( 1 + 4 x ) ^{2} - 8 ( 1 + 4 x ) ( 2 x + 4 x ^{2} )}{( ( 1 + 4 x ) ^{2} ) ^{2}} : \frac{2}{( 1 + 4 x ) ^{3}}

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