derivative of (x^3+x^2-2(7+x-x^4))

解

\frac{d}{d x} ((x^{3} + x^{2} - 2) (7 + x - x^{4}))

- 7 x^{6} - 6 x^{5} + 12 x^{3} + 24 x^{2} + 14 x - 2

解答ステップ

\frac{d}{d x} ((x^{3} + x^{2} - 2) (7 + x - x^{4}))

積の法則を適用:

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

f = x^{3} + x^{2} - 2, g = 7 + x - x^{4}

= \frac{d}{d x} (x^{3} + x^{2} - 2) (7 + x - x^{4}) + \frac{d}{d x} (7 + x - x^{4}) (x^{3} + x^{2} - 2)

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

\frac{d}{d x} (7 + x - x^{4}) = - 4 x^{3} + 1

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

簡素化

(3 x^{2} + 2 x) (7 + x - x^{4}) + (- 4 x^{3} + 1) (x^{3} + x^{2} - 2) : - 7 x^{6} - 6 x^{5} + 12 x^{3} + 24 x^{2} + 14 x - 2

= - 7 x^{6} - 6 x^{5} + 12 x^{3} + 24 x^{2} + 14 x - 2