derivative f(x)=(3x^2-5x)e^x

解

導関数

f (x) = (3 x^{2} - 5 x) e^{x}

3 e^{x} x^{2} + e^{x} x - 5 e^{x}

解答ステップ

\frac{d}{d x} ((3 x^{2} - 5 x) e^{x})

積の法則を適用:

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

f = 3 x^{2} - 5 x, g = e^{x}

= \frac{d}{d x} (3 x^{2} - 5 x) e^{x} + \frac{d}{d x} (e^{x}) (3 x^{2} - 5 x)

\frac{d}{d x} (3 x^{2} - 5 x) = 6 x - 5

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

= (6 x - 5) e^{x} + e^{x} (3 x^{2} - 5 x)

簡素化

(6 x - 5) e^{x} + e^{x} (3 x^{2} - 5 x) : 3 e^{x} x^{2} + e^{x} x - 5 e^{x}

= 3 e^{x} x^{2} + e^{x} x - 5 e^{x}