derivative (t-sqrt(t))/(t^{1/6)}

解

導関数

\frac{t - t}{t ^{\frac{1}{6}}}

\frac{t - 1}{3 t ^{\frac{2}{3}}} + \frac{1}{2 t ^{\frac{1}{6}}}

解答ステップ

\frac{d}{d t} (\frac{t - t}{t ^{\frac{1}{6}}})

簡素化

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

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

積の法則を適用:

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

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

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

\frac{d}{d t} (t^{\frac{1}{3}}) = \frac{1}{3 t ^{\frac{2}{3}}}

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

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

簡素化

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

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