derivative of x^{4/5}*(x-4^2)

解答

\frac{d}{d x} (x^{\frac{4}{5}} \cdot (x - 4)^{2})

\frac{14 x ^{2} - 72 x + 64}{5 x ^{\frac{1}{5}}}

求解步骤

\frac{d}{d x} (x^{\frac{4}{5}} (x - 4)^{2})

使用微分乘法定则:

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

f = x^{\frac{4}{5}}, g = (x - 4)^{2}

= \frac{d}{d x} (x^{\frac{4}{5}}) (x - 4)^{2} + \frac{d}{d x} ((x - 4)^{2}) x^{\frac{4}{5}}

\frac{d}{d x} (x^{\frac{4}{5}}) = \frac{4}{5 x ^{\frac{1}{5}}}

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

= \frac{4}{5 x ^{\frac{1}{5}}} (x - 4)^{2} + 2 (x - 4) x^{\frac{4}{5}}

化简

\frac{4}{5 x ^{\frac{1}{5}}} (x - 4)^{2} + 2 (x - 4) x^{\frac{4}{5}} : \frac{14 x ^{2} - 72 x + 64}{5 x ^{\frac{1}{5}}}

= \frac{14 x ^{2} - 72 x + 64}{5 x ^{\frac{1}{5}}}

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