derivative (x+1)/(3x^2+6x+7)

解

導関数

\frac{x + 1}{3 x ^{2} + 6 x + 7}

\frac{- 3 x ^{2} - 6 x + 1}{( 3 x ^{2} + 6 x + 7 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x + 1}{3 x ^{2} + 6 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} ( x + 1 ) ( 3 x ^{2} + 6 x + 7 ) - \frac{d}{d x} ( 3 x ^{2} + 6 x + 7 ) ( x + 1 )}{( 3 x ^{2} + 6 x + 7 ) ^{2}}

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

\frac{d}{d x} (3 x^{2} + 6 x + 7) = 6 x + 6

= \frac{1 \cdot ( 3 x ^{2} + 6 x + 7 ) - ( 6 x + 6 ) ( x + 1 )}{( 3 x ^{2} + 6 x + 7 ) ^{2}}

簡素化

\frac{1 \cdot ( 3 x ^{2} + 6 x + 7 ) - ( 6 x + 6 ) ( x + 1 )}{( 3 x ^{2} + 6 x + 7 ) ^{2}} : \frac{- 3 x ^{2} - 6 x + 1}{( 3 x ^{2} + 6 x + 7 ) ^{2}}

= \frac{- 3 x ^{2} - 6 x + 1}{( 3 x ^{2} + 6 x + 7 ) ^{2}}