derivative f(x)=(9x^2-15x)e^x

解

導関数

f (x) = (9 x^{2} - 15 x) e^{x}

9 e^{x} x^{2} + 3 e^{x} x - 15 e^{x}

解答ステップ

\frac{d}{d x} ((9 x^{2} - 15 x) e^{x})

積の法則を適用:

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

f = 9 x^{2} - 15 x, g = e^{x}

= \frac{d}{d x} (9 x^{2} - 15 x) e^{x} + \frac{d}{d x} (e^{x}) (9 x^{2} - 15 x)

\frac{d}{d x} (9 x^{2} - 15 x) = 18 x - 15

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

= (18 x - 15) e^{x} + e^{x} (9 x^{2} - 15 x)

簡素化

(18 x - 15) e^{x} + e^{x} (9 x^{2} - 15 x) : 9 e^{x} x^{2} + 3 e^{x} x - 15 e^{x}

= 9 e^{x} x^{2} + 3 e^{x} x - 15 e^{x}