derivative of x^3e^{sqrt(x)}

解

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

3 e^{x} x^{2} + \frac{e ^{x} x ^{\frac{5}{2}}}{2}

解答ステップ

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

積の法則を適用:

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

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

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

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

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

= 3 x^{2} e^{x} + \frac{e ^{x}}{2 x} x^{3}

\frac{e ^{x}}{2 x} x^{3} = \frac{e ^{x} x ^{\frac{5}{2}}}{2}

= 3 e^{x} x^{2} + \frac{e ^{x} x ^{\frac{5}{2}}}{2}