derivative (4x^2-5x+9)/(3x+7)

解

導関数

\frac{4 x ^{2} - 5 x + 9}{3 x + 7}

\frac{12 x ^{2} + 56 x - 62}{( 3 x + 7 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{4 x ^{2} - 5 x + 9}{3 x + 7})

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d x} ( 4 x ^{2} - 5 x + 9 ) ( 3 x + 7 ) - \frac{d}{d x} ( 3 x + 7 ) ( 4 x ^{2} - 5 x + 9 )}{( 3 x + 7 ) ^{2}}

\frac{d}{d x} (4 x^{2} - 5 x + 9) = 8 x - 5

\frac{d}{d x} (3 x + 7) = 3

= \frac{( 8 x - 5 ) ( 3 x + 7 ) - 3 ( 4 x ^{2} - 5 x + 9 )}{( 3 x + 7 ) ^{2}}

拡張

(8 x - 5) (3 x + 7) - 3 (4 x^{2} - 5 x + 9) : 12 x^{2} + 56 x - 62

= \frac{12 x ^{2} + 56 x - 62}{( 3 x + 7 ) ^{2}}