derivative (8x^2+18x)/((9+8x)^2)

解

導関数

\frac{8 x ^{2} + 18 x}{( 9 + 8 x ) ^{2}}

\frac{162}{( 9 + 8 x ) ^{3}}

解答ステップ

\frac{d}{d x} (\frac{8 x ^{2} + 18 x}{( 9 + 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} ( 8 x ^{2} + 18 x ) ( 9 + 8 x ) ^{2} - \frac{d}{d x} ( ( 9 + 8 x ) ^{2} ) ( 8 x ^{2} + 18 x )}{( ( 9 + 8 x ) ^{2} ) ^{2}}

\frac{d}{d x} (8 x^{2} + 18 x) = 16 x + 18

\frac{d}{d x} ((9 + 8 x)^{2}) = 16 (9 + 8 x)

= \frac{( 16 x + 18 ) ( 9 + 8 x ) ^{2} - 16 ( 9 + 8 x ) ( 8 x ^{2} + 18 x )}{( ( 9 + 8 x ) ^{2} ) ^{2}}

簡素化

\frac{( 16 x + 18 ) ( 9 + 8 x ) ^{2} - 16 ( 9 + 8 x ) ( 8 x ^{2} + 18 x )}{( ( 9 + 8 x ) ^{2} ) ^{2}} : \frac{162}{( 9 + 8 x ) ^{3}}

= \frac{162}{( 9 + 8 x ) ^{3}}