derivative of ((x+2(x+1))/(2x-1))

解

\frac{d}{d x} (\frac{( x + 2 ) ( x + 1 )}{2 x - 1})

\frac{2 x ^{2} - 2 x - 7}{( 2 x - 1 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{( x + 2 ) ( x + 1 )}{2 x - 1})

商の法則を適用:

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

= \frac{\frac{d}{d x} ( ( x + 2 ) ( x + 1 ) ) ( 2 x - 1 ) - \frac{d}{d x} ( 2 x - 1 ) ( x + 2 ) ( x + 1 )}{( 2 x - 1 ) ^{2}}

\frac{d}{d x} ((x + 2) (x + 1)) = 2 x + 3

\frac{d}{d x} (2 x - 1) = 2

= \frac{( 2 x + 3 ) ( 2 x - 1 ) - 2 ( x + 2 ) ( x + 1 )}{( 2 x - 1 ) ^{2}}

拡張

(2 x + 3) (2 x - 1) - 2 (x + 2) (x + 1) : 2 x^{2} - 2 x - 7

= \frac{2 x ^{2} - 2 x - 7}{( 2 x - 1 ) ^{2}}