derivative of e^x(x^2+5x)

解

\frac{d}{d x} (e^{x} (x^{2} + 5 x))

e^{x} x^{2} + 7 e^{x} x + 5 e^{x}

解答ステップ

\frac{d}{d x} (e^{x} (x^{2} + 5 x))

積の法則を適用:

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

f = e^{x}, g = x^{2} + 5 x

= \frac{d}{d x} (e^{x}) (x^{2} + 5 x) + \frac{d}{d x} (x^{2} + 5 x) e^{x}

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

\frac{d}{d x} (x^{2} + 5 x) = 2 x + 5

= e^{x} (x^{2} + 5 x) + (2 x + 5) e^{x}

簡素化

e^{x} (x^{2} + 5 x) + (2 x + 5) e^{x} : e^{x} x^{2} + 7 e^{x} x + 5 e^{x}

= e^{x} x^{2} + 7 e^{x} x + 5 e^{x}