derivative of x^{3/5}(4-x)

解答

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

\frac{12 - 8 x}{5 x ^{\frac{2}{5}}}

求解步骤

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

使用微分乘法定则:

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

f = x^{\frac{3}{5}}, g = 4 - x

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

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

\frac{d}{d x} (4 - x) = - 1

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

化简

\frac{3}{5 x ^{\frac{2}{5}}} (4 - x) + (- 1) x^{\frac{3}{5}} : \frac{12 - 8 x}{5 x ^{\frac{2}{5}}}

= \frac{12 - 8 x}{5 x ^{\frac{2}{5}}}

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