derivative f(x)=(11x+7)/(3x^4+2x^2)

解

導関数

f (x) = \frac{11 x + 7}{3 x ^{4} + 2 x ^{2}}

- \frac{99 x ^{3} + 84 x ^{2} + 22 x + 28}{x ^{3} ( 3 x ^{2} + 2 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{11 x + 7}{3 x ^{4} + 2 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} ( 11 x + 7 ) ( 3 x ^{4} + 2 x ^{2} ) - \frac{d}{d x} ( 3 x ^{4} + 2 x ^{2} ) ( 11 x + 7 )}{( 3 x ^{4} + 2 x ^{2} ) ^{2}}

\frac{d}{d x} (11 x + 7) = 11

\frac{d}{d x} (3 x^{4} + 2 x^{2}) = 12 x^{3} + 4 x

= \frac{11 ( 3 x ^{4} + 2 x ^{2} ) - ( 12 x ^{3} + 4 x ) ( 11 x + 7 )}{( 3 x ^{4} + 2 x ^{2} ) ^{2}}

簡素化

\frac{11 ( 3 x ^{4} + 2 x ^{2} ) - ( 12 x ^{3} + 4 x ) ( 11 x + 7 )}{( 3 x ^{4} + 2 x ^{2} ) ^{2}} : - \frac{99 x ^{3} + 84 x ^{2} + 22 x + 28}{x ^{3} ( 3 x ^{2} + 2 ) ^{2}}

= - \frac{99 x ^{3} + 84 x ^{2} + 22 x + 28}{x ^{3} ( 3 x ^{2} + 2 ) ^{2}}