derivative (x^3-7x+2)(2x^4-1)

解答

导数

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

14 x^{6} - 70 x^{4} + 16 x^{3} - 3 x^{2} + 7

求解步骤

\frac{d}{d x} ((x^{3} - 7 x + 2) (2 x^{4} - 1))

使用微分乘法定则:

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

f = x^{3} - 7 x + 2, g = 2 x^{4} - 1

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

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

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

= (3 x^{2} - 7) (2 x^{4} - 1) + 8 x^{3} (x^{3} - 7 x + 2)

化简

(3 x^{2} - 7) (2 x^{4} - 1) + 8 x^{3} (x^{3} - 7 x + 2) : 14 x^{6} - 70 x^{4} + 16 x^{3} - 3 x^{2} + 7

= 14 x^{6} - 70 x^{4} + 16 x^{3} - 3 x^{2} + 7

x \to 1 lim (1 - x^{3})

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