derivative f(x)=(3-x)^4(x+5)^3

解答

导数

f (x) = (3 - x)^{4} (x + 5)^{3}

- 4 (3 - x)^{3} (x + 5)^{3} + 3 (x + 5)^{2} (3 - x)^{4}

求解步骤

\frac{d}{d x} ((3 - x)^{4} (x + 5)^{3})

使用微分乘法定则:

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

f = (3 - x)^{4}, g = (x + 5)^{3}

= \frac{d}{d x} ((3 - x)^{4}) (x + 5)^{3} + \frac{d}{d x} ((x + 5)^{3}) (3 - x)^{4}

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

\frac{d}{d x} ((x + 5)^{3}) = 3 (x + 5)^{2}

= (- 4 (3 - x)^{3}) (x + 5)^{3} + 3 (x + 5)^{2} (3 - x)^{4}

化简

= - 4 (3 - x)^{3} (x + 5)^{3} + 3 (x + 5)^{2} (3 - x)^{4}

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