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

解

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

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

解答ステップ

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

\frac{d}{d x} (x e^{x}) = e^{x} + e^{x} x

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

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

簡素化

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

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