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

解

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

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

解答ステップ

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

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

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

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

拡張

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

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