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

解

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

\frac{12 x ^{2} - 18 x - 6}{( 4 x - 3 ) ^{2}}

解答ステップ

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

商の法則を適用:

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

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

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

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

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

拡張

(6 x + 2) (4 x - 3) - 4 (3 x^{2} + 2 x) : 12 x^{2} - 18 x - 6

= \frac{12 x ^{2} - 18 x - 6}{( 4 x - 3 ) ^{2}}