derivative of (x^5+2x(x^3+x^2+x+7))

解

\frac{d}{d x} ((x^{5} + 2 x) (x^{3} + x^{2} + x + 7))

8 x^{7} + 7 x^{6} + 6 x^{5} + 35 x^{4} + 8 x^{3} + 6 x^{2} + 4 x + 14

解答ステップ

\frac{d}{d x} ((x^{5} + 2 x) (x^{3} + x^{2} + x + 7))

積の法則を適用:

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

f = x^{5} + 2 x, g = x^{3} + x^{2} + x + 7

= \frac{d}{d x} (x^{5} + 2 x) (x^{3} + x^{2} + x + 7) + \frac{d}{d x} (x^{3} + x^{2} + x + 7) (x^{5} + 2 x)

\frac{d}{d x} (x^{5} + 2 x) = 5 x^{4} + 2

\frac{d}{d x} (x^{3} + x^{2} + x + 7) = 3 x^{2} + 2 x + 1

= (5 x^{4} + 2) (x^{3} + x^{2} + x + 7) + (3 x^{2} + 2 x + 1) (x^{5} + 2 x)

簡素化

(5 x^{4} + 2) (x^{3} + x^{2} + x + 7) + (3 x^{2} + 2 x + 1) (x^{5} + 2 x) : 8 x^{7} + 7 x^{6} + 6 x^{5} + 35 x^{4} + 8 x^{3} + 6 x^{2} + 4 x + 14

= 8 x^{7} + 7 x^{6} + 6 x^{5} + 35 x^{4} + 8 x^{3} + 6 x^{2} + 4 x + 14