derivative of (6x+x^2/((3+x)^2))

解

\frac{d}{d x} (\frac{6 x + x ^{2}}{( 3 + x ) ^{2}})

\frac{18}{( 3 + x ) ^{3}}

解答ステップ

\frac{d}{d x} (\frac{6 x + x ^{2}}{( 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} ( 6 x + x ^{2} ) ( 3 + x ) ^{2} - \frac{d}{d x} ( ( 3 + x ) ^{2} ) ( 6 x + x ^{2} )}{( ( 3 + x ) ^{2} ) ^{2}}

\frac{d}{d x} (6 x + x^{2}) = 6 + 2 x

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

= \frac{( 6 + 2 x ) ( 3 + x ) ^{2} - 2 ( 3 + x ) ( 6 x + x ^{2} )}{( ( 3 + x ) ^{2} ) ^{2}}

簡素化

\frac{( 6 + 2 x ) ( 3 + x ) ^{2} - 2 ( 3 + x ) ( 6 x + x ^{2} )}{( ( 3 + x ) ^{2} ) ^{2}} : \frac{18}{( 3 + x ) ^{3}}

= \frac{18}{( 3 + x ) ^{3}}