derivative (t-sqrt(t))/(t^{1/5)}

解

導関数

\frac{t - t}{t ^{\frac{1}{5}}}

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

解答ステップ

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

簡素化

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

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

積の法則を適用:

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

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

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

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

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

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

簡素化

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

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