derivative f(x)=(10x+8x^2)/((5+8x)^2)

解

導関数

f (x) = \frac{10 x + 8 x ^{2}}{( 5 + 8 x ) ^{2}}

\frac{50}{( 5 + 8 x ) ^{3}}

解答ステップ

\frac{d}{d x} (\frac{10 x + 8 x ^{2}}{( 5 + 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} ( 10 x + 8 x ^{2} ) ( 5 + 8 x ) ^{2} - \frac{d}{d x} ( ( 5 + 8 x ) ^{2} ) ( 10 x + 8 x ^{2} )}{( ( 5 + 8 x ) ^{2} ) ^{2}}

\frac{d}{d x} (10 x + 8 x^{2}) = 10 + 16 x

\frac{d}{d x} ((5 + 8 x)^{2}) = 16 (5 + 8 x)

= \frac{( 10 + 16 x ) ( 5 + 8 x ) ^{2} - 16 ( 5 + 8 x ) ( 10 x + 8 x ^{2} )}{( ( 5 + 8 x ) ^{2} ) ^{2}}

簡素化

\frac{( 10 + 16 x ) ( 5 + 8 x ) ^{2} - 16 ( 5 + 8 x ) ( 10 x + 8 x ^{2} )}{( ( 5 + 8 x ) ^{2} ) ^{2}} : \frac{50}{( 5 + 8 x ) ^{3}}

= \frac{50}{( 5 + 8 x ) ^{3}}