derivative y=(x+1)^2(x^2+2x)

解

導関数

y = (x + 1)^{2} (x^{2} + 2 x)

4 x^{3} + 12 x^{2} + 10 x + 2

解答ステップ

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

積の法則を適用:

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

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

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

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

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

= 2 (x + 1) (x^{2} + 2 x) + (2 x + 2) (x + 1)^{2}

簡素化

2 (x + 1) (x^{2} + 2 x) + (2 x + 2) (x + 1)^{2} : 4 x^{3} + 12 x^{2} + 10 x + 2

= 4 x^{3} + 12 x^{2} + 10 x + 2