derivative of x^6e^{2x^3}

解

\frac{d}{d x} (x^{6} e^{2 x^{3}})

6 e^{2 x^{3}} x^{5} + 6 e^{2 x^{3}} x^{8}

解答ステップ

\frac{d}{d x} (x^{6} e^{2 x^{3}})

積の法則を適用:

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

f = x^{6}, g = e^{2 x^{3}}

= \frac{d}{d x} (x^{6}) e^{2 x^{3}} + \frac{d}{d x} (e^{2 x^{3}}) x^{6}

\frac{d}{d x} (x^{6}) = 6 x^{5}

\frac{d}{d x} (e^{2 x^{3}}) = e^{2 x^{3}} \cdot 6 x^{2}

= 6 x^{5} e^{2 x^{3}} + e^{2 x^{3}} \cdot 6 x^{2} x^{6}

e^{2 x^{3}} \cdot 6 x^{2} x^{6} = 6 e^{2 x^{3}} x^{8}

= 6 e^{2 x^{3}} x^{5} + 6 e^{2 x^{3}} x^{8}