derivative (9z^4+3z^2+1)(3z^3-z)

解答

导数

(9 z^{4} + 3 z^{2} + 1) (3 z^{3} - z)

189 z^{6} - 1

求解步骤

\frac{d}{d z} ((9 z^{4} + 3 z^{2} + 1) (3 z^{3} - z))

使用微分乘法定则:

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

f = 9 z^{4} + 3 z^{2} + 1, g = 3 z^{3} - z

= \frac{d}{d z} (9 z^{4} + 3 z^{2} + 1) (3 z^{3} - z) + \frac{d}{d z} (3 z^{3} - z) (9 z^{4} + 3 z^{2} + 1)

\frac{d}{d z} (9 z^{4} + 3 z^{2} + 1) = 36 z^{3} + 6 z

\frac{d}{d z} (3 z^{3} - z) = 9 z^{2} - 1

= (36 z^{3} + 6 z) (3 z^{3} - z) + (9 z^{2} - 1) (9 z^{4} + 3 z^{2} + 1)

化简

(36 z^{3} + 6 z) (3 z^{3} - z) + (9 z^{2} - 1) (9 z^{4} + 3 z^{2} + 1) : 189 z^{6} - 1

= 189 z^{6} - 1

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