derivative of (x^2/(x^4+1))

解

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

\frac{- 2 x ^{5} + 2 x}{( x ^{4} + 1 ) ^{2}}

解答ステップ

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

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

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

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

簡素化

\frac{2 x ( x ^{4} + 1 ) - 4 x ^{3} x ^{2}}{( x ^{4} + 1 ) ^{2}} : \frac{- 2 x ^{5} + 2 x}{( x ^{4} + 1 ) ^{2}}

= \frac{- 2 x ^{5} + 2 x}{( x ^{4} + 1 ) ^{2}}