derivative (18x+4x^2)/((9+4x)^2)

解

導関数

\frac{18 x + 4 x ^{2}}{( 9 + 4 x ) ^{2}}

\frac{162}{( 9 + 4 x ) ^{3}}

解答ステップ

\frac{d}{d x} (\frac{18 x + 4 x ^{2}}{( 9 + 4 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} ( 18 x + 4 x ^{2} ) ( 9 + 4 x ) ^{2} - \frac{d}{d x} ( ( 9 + 4 x ) ^{2} ) ( 18 x + 4 x ^{2} )}{( ( 9 + 4 x ) ^{2} ) ^{2}}

\frac{d}{d x} (18 x + 4 x^{2}) = 18 + 8 x

\frac{d}{d x} ((9 + 4 x)^{2}) = 8 (9 + 4 x)

= \frac{( 18 + 8 x ) ( 9 + 4 x ) ^{2} - 8 ( 9 + 4 x ) ( 18 x + 4 x ^{2} )}{( ( 9 + 4 x ) ^{2} ) ^{2}}

簡素化

\frac{( 18 + 8 x ) ( 9 + 4 x ) ^{2} - 8 ( 9 + 4 x ) ( 18 x + 4 x ^{2} )}{( ( 9 + 4 x ) ^{2} ) ^{2}} : \frac{162}{( 9 + 4 x ) ^{3}}

= \frac{162}{( 9 + 4 x ) ^{3}}