derivative of (5x^2-9x+7/(3x+2))

解

\frac{d}{d x} (\frac{5 x ^{2} - 9 x + 7}{3 x + 2})

\frac{15 x ^{2} + 20 x - 39}{( 3 x + 2 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{5 x ^{2} - 9 x + 7}{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} ( 5 x ^{2} - 9 x + 7 ) ( 3 x + 2 ) - \frac{d}{d x} ( 3 x + 2 ) ( 5 x ^{2} - 9 x + 7 )}{( 3 x + 2 ) ^{2}}

\frac{d}{d x} (5 x^{2} - 9 x + 7) = 10 x - 9

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

= \frac{( 10 x - 9 ) ( 3 x + 2 ) - 3 ( 5 x ^{2} - 9 x + 7 )}{( 3 x + 2 ) ^{2}}

拡張

(10 x - 9) (3 x + 2) - 3 (5 x^{2} - 9 x + 7) : 15 x^{2} + 20 x - 39

= \frac{15 x ^{2} + 20 x - 39}{( 3 x + 2 ) ^{2}}