derivative 2x^{1/5}(x^3-8)

解答

导数

2 x^{\frac{1}{5}} (x^{3} - 8)

\frac{2 ( 16 x ^{3} - 8 )}{5 x ^{\frac{4}{5}}}

求解步骤

\frac{d}{d x} (2 x^{\frac{1}{5}} (x^{3} - 8))

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \frac{d}{d x} (x^{\frac{1}{5}} (x^{3} - 8))

使用微分乘法定则:

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

f = x^{\frac{1}{5}}, g = x^{3} - 8

= 2 (\frac{d}{d x} (x^{\frac{1}{5}}) (x^{3} - 8) + \frac{d}{d x} (x^{3} - 8) x^{\frac{1}{5}})

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

\frac{d}{d x} (x^{3} - 8) = 3 x^{2}

= 2 (\frac{1}{5 x ^{\frac{4}{5}}} (x^{3} - 8) + 3 x^{2} x^{\frac{1}{5}})

化简

2 (\frac{1}{5 x ^{\frac{4}{5}}} (x^{3} - 8) + 3 x^{2} x^{\frac{1}{5}}) : \frac{2 ( 16 x ^{3} - 8 )}{5 x ^{\frac{4}{5}}}

= \frac{2 ( 16 x ^{3} - 8 )}{5 x ^{\frac{4}{5}}}

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