derivative of ((x-sqrt(x))/(x^{1/5)})

解

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

\frac{3 ( x - 1 )}{10 x ^{\frac{7}{10}}} + \frac{1}{2 x ^{\frac{1}{5}}}

解答ステップ

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

簡素化

\frac{( x - x )}{x ^{\frac{1}{5}}} : x^{\frac{3}{10}} (x - 1)

= \frac{d}{d x} (x^{\frac{3}{10}} (x - 1))

積の法則を適用:

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

f = x^{\frac{3}{10}}, g = x - 1

= \frac{d}{d x} (x^{\frac{3}{10}}) (x - 1) + \frac{d}{d x} (x - 1) x^{\frac{3}{10}}

\frac{d}{d x} (x^{\frac{3}{10}}) = \frac{3}{10 x ^{\frac{7}{10}}}

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

= \frac{3}{10 x ^{\frac{7}{10}}} (x - 1) + \frac{1}{2 x ^{\frac{1}{2}}} x^{\frac{3}{10}}

簡素化

\frac{3}{10 x ^{\frac{7}{10}}} (x - 1) + \frac{1}{2 x ^{\frac{1}{2}}} x^{\frac{3}{10}} : \frac{3 ( x - 1 )}{10 x ^{\frac{7}{10}}} + \frac{1}{2 x ^{\frac{1}{5}}}

= \frac{3 ( x - 1 )}{10 x ^{\frac{7}{10}}} + \frac{1}{2 x ^{\frac{1}{5}}}