derivative (2x^2+x-3)/(-3x^2-4x+2)

解

導関数

\frac{2 x ^{2} + x - 3}{- 3 x ^{2} - 4 x + 2}

\frac{- 5 x ^{2} - 10 x - 10}{( - 3 x ^{2} - 4 x + 2 ) ^{2}}

解答ステップ

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

\frac{d}{d x} (2 x^{2} + x - 3) = 4 x + 1

\frac{d}{d x} (- 3 x^{2} - 4 x + 2) = - 6 x - 4

= \frac{( 4 x + 1 ) ( - 3 x ^{2} - 4 x + 2 ) - ( - 6 x - 4 ) ( 2 x ^{2} + x - 3 )}{( - 3 x ^{2} - 4 x + 2 ) ^{2}}

拡張

(4 x + 1) (- 3 x^{2} - 4 x + 2) - (- 6 x - 4) (2 x^{2} + x - 3) : - 5 x^{2} - 10 x - 10

= \frac{- 5 x ^{2} - 10 x - 10}{( - 3 x ^{2} - 4 x + 2 ) ^{2}}