derivative f(x)=(8x^3-3x+2)/(3x-x^4)

解

導関数

f (x) = \frac{8 x ^{3} - 3 x + 2}{3 x - x ^{4}}

\frac{8 x ^{6} - 9 x ^{4} + 56 x ^{3} - 6}{( 3 x - x ^{4} ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{8 x ^{3} - 3 x + 2}{3 x - x ^{4}})

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 ^{3} - 3 x + 2 ) ( 3 x - x ^{4} ) - \frac{d}{d x} ( 3 x - x ^{4} ) ( 8 x ^{3} - 3 x + 2 )}{( 3 x - x ^{4} ) ^{2}}

\frac{d}{d x} (8 x^{3} - 3 x + 2) = 24 x^{2} - 3

\frac{d}{d x} (3 x - x^{4}) = 3 - 4 x^{3}

= \frac{( 24 x ^{2} - 3 ) ( 3 x - x ^{4} ) - ( 3 - 4 x ^{3} ) ( 8 x ^{3} - 3 x + 2 )}{( 3 x - x ^{4} ) ^{2}}

拡張

(24 x^{2} - 3) (3 x - x^{4}) - (3 - 4 x^{3}) (8 x^{3} - 3 x + 2) : 8 x^{6} - 9 x^{4} + 56 x^{3} - 6

= \frac{8 x ^{6} - 9 x ^{4} + 56 x ^{3} - 6}{( 3 x - x ^{4} ) ^{2}}