derivative (x^2-x+1)e^3x

解

導関数

(x^{2} - x + 1) e^{3} x

e^{3} (3 x^{2} - 2 x + 1)

解答ステップ

\frac{d}{d x} ((x^{2} - x + 1) e^{3} x)

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= e^{3} \frac{d}{d x} ((x^{2} - x + 1) x)

積の法則を適用:

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

f = x^{2} - x + 1, g = x

= e^{3} (\frac{d}{d x} (x^{2} - x + 1) x + \frac{d x}{d x} (x^{2} - x + 1))

\frac{d}{d x} (x^{2} - x + 1) = 2 x - 1

\frac{d x}{d x} = 1

= e^{3} ((2 x - 1) x + 1 \cdot (x^{2} - x + 1))

簡素化

e^{3} ((2 x - 1) x + 1 \cdot (x^{2} - x + 1)) : e^{3} (3 x^{2} - 2 x + 1)

= e^{3} (3 x^{2} - 2 x + 1)