derivative of (14x+8x^2/((7+8x)^2))

解

\frac{d}{d x} (\frac{14 x + 8 x ^{2}}{( 7 + 8 x ) ^{2}})

\frac{98}{( 7 + 8 x ) ^{3}}

解答ステップ

\frac{d}{d x} (\frac{14 x + 8 x ^{2}}{( 7 + 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} ( 14 x + 8 x ^{2} ) ( 7 + 8 x ) ^{2} - \frac{d}{d x} ( ( 7 + 8 x ) ^{2} ) ( 14 x + 8 x ^{2} )}{( ( 7 + 8 x ) ^{2} ) ^{2}}

\frac{d}{d x} (14 x + 8 x^{2}) = 14 + 16 x

\frac{d}{d x} ((7 + 8 x)^{2}) = 16 (7 + 8 x)

= \frac{( 14 + 16 x ) ( 7 + 8 x ) ^{2} - 16 ( 7 + 8 x ) ( 14 x + 8 x ^{2} )}{( ( 7 + 8 x ) ^{2} ) ^{2}}

簡素化

\frac{( 14 + 16 x ) ( 7 + 8 x ) ^{2} - 16 ( 7 + 8 x ) ( 14 x + 8 x ^{2} )}{( ( 7 + 8 x ) ^{2} ) ^{2}} : \frac{98}{( 7 + 8 x ) ^{3}}

= \frac{98}{( 7 + 8 x ) ^{3}}