derivative f(x)=(2x^3+6x)(4x^2-7x+11)

解答

导数

f (x) = (2 x^{3} + 6 x) (4 x^{2} - 7 x + 11)

40 x^{4} - 56 x^{3} + 138 x^{2} - 84 x + 66

求解步骤

\frac{d}{d x} ((2 x^{3} + 6 x) (4 x^{2} - 7 x + 11))

使用微分乘法定则:

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

f = 2 x^{3} + 6 x, g = 4 x^{2} - 7 x + 11

= \frac{d}{d x} (2 x^{3} + 6 x) (4 x^{2} - 7 x + 11) + \frac{d}{d x} (4 x^{2} - 7 x + 11) (2 x^{3} + 6 x)

\frac{d}{d x} (2 x^{3} + 6 x) = 6 x^{2} + 6

\frac{d}{d x} (4 x^{2} - 7 x + 11) = 8 x - 7

= (6 x^{2} + 6) (4 x^{2} - 7 x + 11) + (8 x - 7) (2 x^{3} + 6 x)

化简

(6 x^{2} + 6) (4 x^{2} - 7 x + 11) + (8 x - 7) (2 x^{3} + 6 x) : 40 x^{4} - 56 x^{3} + 138 x^{2} - 84 x + 66

= 40 x^{4} - 56 x^{3} + 138 x^{2} - 84 x + 66

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

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