derivative of (sqrt(x)+1/(x^2+1))

解

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

\frac{- 3 x ^{2} - 4 x x + 1}{2 x ( x ^{2} + 1 ) ^{2}}

解答ステップ

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

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

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

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

簡素化

\frac{\frac{1}{2 x} ( x ^{2} + 1 ) - 2 x ( x + 1 )}{( x ^{2} + 1 ) ^{2}} : \frac{- 3 x ^{2} - 4 x x + 1}{2 x ( x ^{2} + 1 ) ^{2}}

= \frac{- 3 x ^{2} - 4 x x + 1}{2 x ( x ^{2} + 1 ) ^{2}}