derivative of (x-sqrt(x)/(x^{1/4)})

解答

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

\frac{x - 1}{4 x ^{\frac{3}{4}}} + \frac{1}{2 x ^{\frac{1}{4}}}

求解步骤

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

化简

\frac{x - x}{x ^{\frac{1}{4}}} : x^{\frac{1}{4}} (x - 1)

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

使用微分乘法定则:

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

f = x^{\frac{1}{4}}, g = x - 1

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

\frac{d}{d x} (x^{\frac{1}{4}}) = \frac{1}{4 x ^{\frac{3}{4}}}

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

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

化简

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

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

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