derivative (t-sqrt(t))/(t^{1/4)}

解

導関数

\frac{t - t}{t ^{\frac{1}{4}}}

\frac{t - 1}{4 t ^{\frac{3}{4}}} + \frac{1}{2 t ^{\frac{1}{4}}}

解答ステップ

\frac{d}{d t} (\frac{t - t}{t ^{\frac{1}{4}}})

簡素化

\frac{t - t}{t ^{\frac{1}{4}}} : t^{\frac{1}{4}} (t - 1)

= \frac{d}{d t} (t^{\frac{1}{4}} (t - 1))

積の法則を適用:

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

f = t^{\frac{1}{4}}, g = t - 1

= \frac{d}{d t} (t^{\frac{1}{4}}) (t - 1) + \frac{d}{d t} (t - 1) t^{\frac{1}{4}}

\frac{d}{d t} (t^{\frac{1}{4}}) = \frac{1}{4 t ^{\frac{3}{4}}}

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

= \frac{1}{4 t ^{\frac{3}{4}}} (t - 1) + \frac{1}{2 t ^{\frac{1}{2}}} t^{\frac{1}{4}}

簡素化

\frac{1}{4 t ^{\frac{3}{4}}} (t - 1) + \frac{1}{2 t ^{\frac{1}{2}}} t^{\frac{1}{4}} : \frac{t - 1}{4 t ^{\frac{3}{4}}} + \frac{1}{2 t ^{\frac{1}{4}}}

= \frac{t - 1}{4 t ^{\frac{3}{4}}} + \frac{1}{2 t ^{\frac{1}{4}}}