derivative of (6x+8x^2/((3+8x)^2))

解

\frac{d}{d x} (\frac{6 x + 8 x ^{2}}{( 3 + 8 x ) ^{2}})

\frac{18}{( 3 + 8 x ) ^{3}}

解答ステップ

\frac{d}{d x} (\frac{6 x + 8 x ^{2}}{( 3 + 8 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 + 8 x ^{2} ) ( 3 + 8 x ) ^{2} - \frac{d}{d x} ( ( 3 + 8 x ) ^{2} ) ( 6 x + 8 x ^{2} )}{( ( 3 + 8 x ) ^{2} ) ^{2}}

\frac{d}{d x} (6 x + 8 x^{2}) = 6 + 16 x

\frac{d}{d x} ((3 + 8 x)^{2}) = 16 (3 + 8 x)

= \frac{( 6 + 16 x ) ( 3 + 8 x ) ^{2} - 16 ( 3 + 8 x ) ( 6 x + 8 x ^{2} )}{( ( 3 + 8 x ) ^{2} ) ^{2}}

簡素化

\frac{( 6 + 16 x ) ( 3 + 8 x ) ^{2} - 16 ( 3 + 8 x ) ( 6 x + 8 x ^{2} )}{( ( 3 + 8 x ) ^{2} ) ^{2}} : \frac{18}{( 3 + 8 x ) ^{3}}

= \frac{18}{( 3 + 8 x ) ^{3}}