derivative of xe^{(x^2/2})

解

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

e^{\frac{x ^{2}}{2}} + e^{\frac{x ^{2}}{2}} x^{2}

解答ステップ

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

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = e^{\frac{x ^{2}}{2}}

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

\frac{d x}{d x} = 1

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

= 1 \cdot e^{\frac{x ^{2}}{2}} + e^{\frac{x ^{2}}{2}} xx

簡素化

1 \cdot e^{\frac{x ^{2}}{2}} + e^{\frac{x ^{2}}{2}} xx : e^{\frac{x ^{2}}{2}} + e^{\frac{x ^{2}}{2}} x^{2}

= e^{\frac{x ^{2}}{2}} + e^{\frac{x ^{2}}{2}} x^{2}