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

解

導関数

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

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

解答ステップ

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

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

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

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

拡張

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

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