derivative f(x)=(x^2+1)(x^3+1)

解

導関数

f (x) = (x^{2} + 1) (x^{3} + 1)

5 x^{4} + 3 x^{2} + 2 x

解答ステップ

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

積の法則を適用:

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

f = x^{2} + 1, g = x^{3} + 1

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

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

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

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

簡素化

2 x (x^{3} + 1) + 3 x^{2} (x^{2} + 1) : 5 x^{4} + 3 x^{2} + 2 x

= 5 x^{4} + 3 x^{2} + 2 x