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

解答

导数

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

- 24 x (2 - 3 x^{2})^{3} (x^{7} + 3)^{3} + 21 x^{6} (x^{7} + 3)^{2} (2 - 3 x^{2})^{4}

求解步骤

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

使用微分乘法定则:

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

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

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

\frac{d}{d x} ((2 - 3 x^{2})^{4}) = - 24 x (2 - 3 x^{2})^{3}

\frac{d}{d x} ((x^{7} + 3)^{3}) = 21 x^{6} (x^{7} + 3)^{2}

= (- 24 x (2 - 3 x^{2})^{3}) (x^{7} + 3)^{3} + 21 x^{6} (x^{7} + 3)^{2} (2 - 3 x^{2})^{4}

化简

= - 24 x (2 - 3 x^{2})^{3} (x^{7} + 3)^{3} + 21 x^{6} (x^{7} + 3)^{2} (2 - 3 x^{2})^{4}

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