derivative of (x-sqrt(x)/(x^{1/3)})

解

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

\frac{x - 1}{6 x ^{\frac{5}{6}}} + \frac{1}{2 x ^{\frac{1}{3}}}

解答ステップ

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

簡素化

\frac{x - x}{x ^{\frac{1}{3}}} : x^{\frac{1}{6}} (x - 1)

= \frac{d}{d x} (x^{\frac{1}{6}} (x - 1))

積の法則を適用:

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

f = x^{\frac{1}{6}}, g = x - 1

= \frac{d}{d x} (x^{\frac{1}{6}}) (x - 1) + \frac{d}{d x} (x - 1) x^{\frac{1}{6}}

\frac{d}{d x} (x^{\frac{1}{6}}) = \frac{1}{6 x ^{\frac{5}{6}}}

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

= \frac{1}{6 x ^{\frac{5}{6}}} (x - 1) + \frac{1}{2 x ^{\frac{1}{2}}} x^{\frac{1}{6}}

簡素化

\frac{1}{6 x ^{\frac{5}{6}}} (x - 1) + \frac{1}{2 x ^{\frac{1}{2}}} x^{\frac{1}{6}} : \frac{x - 1}{6 x ^{\frac{5}{6}}} + \frac{1}{2 x ^{\frac{1}{3}}}

= \frac{x - 1}{6 x ^{\frac{5}{6}}} + \frac{1}{2 x ^{\frac{1}{3}}}