derivative f(x)=(8x+x^2)/((4+x)^2)

解

導関数

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

\frac{32}{( 4 + x ) ^{3}}

解答ステップ

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

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

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

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

簡素化

\frac{( 8 + 2 x ) ( 4 + x ) ^{2} - 2 ( 4 + x ) ( 8 x + x ^{2} )}{( ( 4 + x ) ^{2} ) ^{2}} : \frac{32}{( 4 + x ) ^{3}}

= \frac{32}{( 4 + x ) ^{3}}