derivative of xe^{x^5}

解

\frac{d}{d x} (x e^{x^{5}})

e^{x^{5}} + 5 e^{x^{5}} x^{5}

解答ステップ

\frac{d}{d x} (x e^{x^{5}})

積の法則を適用:

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

f = x, g = e^{x^{5}}

= \frac{d x}{d x} e^{x^{5}} + \frac{d}{d x} (e^{x^{5}}) x

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

\frac{d}{d x} (e^{x^{5}}) = e^{x^{5}} \cdot 5 x^{4}

= 1 \cdot e^{x^{5}} + e^{x^{5}} \cdot 5 x^{4} x

簡素化

1 \cdot e^{x^{5}} + e^{x^{5}} \cdot 5 x^{4} x : e^{x^{5}} + 5 e^{x^{5}} x^{5}

= e^{x^{5}} + 5 e^{x^{5}} x^{5}