derivative of (x-sqrt(x)/(x^{1/9)})

解

\frac{d}{d x} (\frac{x - x}{x ^{\frac{1}{9}}})

\frac{7 ( x - 1 )}{18 x ^{\frac{11}{18}}} + \frac{1}{2 x ^{\frac{1}{9}}}

解答ステップ

\frac{d}{d x} (\frac{x - x}{x ^{\frac{1}{9}}})

簡素化

\frac{x - x}{x ^{\frac{1}{9}}} : x^{\frac{7}{18}} (x - 1)

= \frac{d}{d x} (x^{\frac{7}{18}} (x - 1))

積の法則を適用:

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

f = x^{\frac{7}{18}}, g = x - 1

= \frac{d}{d x} (x^{\frac{7}{18}}) (x - 1) + \frac{d}{d x} (x - 1) x^{\frac{7}{18}}

\frac{d}{d x} (x^{\frac{7}{18}}) = \frac{7}{18 x ^{\frac{11}{18}}}

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

= \frac{7}{18 x ^{\frac{11}{18}}} (x - 1) + \frac{1}{2 x ^{\frac{1}{2}}} x^{\frac{7}{18}}

簡素化

\frac{7}{18 x ^{\frac{11}{18}}} (x - 1) + \frac{1}{2 x ^{\frac{1}{2}}} x^{\frac{7}{18}} : \frac{7 ( x - 1 )}{18 x ^{\frac{11}{18}}} + \frac{1}{2 x ^{\frac{1}{9}}}

= \frac{7 ( x - 1 )}{18 x ^{\frac{11}{18}}} + \frac{1}{2 x ^{\frac{1}{9}}}