derivative of axe^{bx^2}

解

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

a (e^{b x^{2}} + 2 b e^{b x^{2}} x^{2})

解答ステップ

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

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

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

積の法則を適用:

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

f = x, g = e^{b x^{2}}

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

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

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

= a (1 \cdot e^{b x^{2}} + e^{b x^{2}} \cdot 2 b xx)

簡素化

a (1 \cdot e^{b x^{2}} + e^{b x^{2}} \cdot 2 b xx) : a (e^{b x^{2}} + 2 b e^{b x^{2}} x^{2})

= a (e^{b x^{2}} + 2 b e^{b x^{2}} x^{2})