derivative e^x(x^2+1)(x+4)

解

導関数

e^{x} (x^{2} + 1) (x + 4)

e^{x} x^{3} + 7 e^{x} x^{2} + 9 e^{x} x + 5 e^{x}

解答ステップ

\frac{d}{d x} (e^{x} (x^{2} + 1) (x + 4))

積の法則を適用:

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

f = e^{x}, g = (x^{2} + 1) (x + 4)

= \frac{d}{d x} (e^{x}) (x^{2} + 1) (x + 4) + \frac{d}{d x} ((x^{2} + 1) (x + 4)) e^{x}

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

\frac{d}{d x} ((x^{2} + 1) (x + 4)) = 3 x^{2} + 8 x + 1

= e^{x} (x^{2} + 1) (x + 4) + (3 x^{2} + 8 x + 1) e^{x}

簡素化

e^{x} (x^{2} + 1) (x + 4) + (3 x^{2} + 8 x + 1) e^{x} : e^{x} x^{3} + 7 e^{x} x^{2} + 9 e^{x} x + 5 e^{x}

= e^{x} x^{3} + 7 e^{x} x^{2} + 9 e^{x} x + 5 e^{x}