derivative of (8x^3-6x^2/(x^2+3x-2))

解

\frac{d}{d x} (\frac{8 x ^{3} - 6 x ^{2}}{x ^{2} + 3 x - 2})

\frac{8 x ^{4} + 48 x ^{3} - 66 x ^{2} + 24 x}{( x ^{2} + 3 x - 2 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{8 x ^{3} - 6 x ^{2}}{x ^{2} + 3 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 ^{3} - 6 x ^{2} ) ( x ^{2} + 3 x - 2 ) - \frac{d}{d x} ( x ^{2} + 3 x - 2 ) ( 8 x ^{3} - 6 x ^{2} )}{( x ^{2} + 3 x - 2 ) ^{2}}

\frac{d}{d x} (8 x^{3} - 6 x^{2}) = 24 x^{2} - 12 x

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

= \frac{( 24 x ^{2} - 12 x ) ( x ^{2} + 3 x - 2 ) - ( 2 x + 3 ) ( 8 x ^{3} - 6 x ^{2} )}{( x ^{2} + 3 x - 2 ) ^{2}}

拡張

(24 x^{2} - 12 x) (x^{2} + 3 x - 2) - (2 x + 3) (8 x^{3} - 6 x^{2}) : 8 x^{4} + 48 x^{3} - 66 x^{2} + 24 x

= \frac{8 x ^{4} + 48 x ^{3} - 66 x ^{2} + 24 x}{( x ^{2} + 3 x - 2 ) ^{2}}