derivative (\sqrt[3]{t})/(t-3)

解

導関数

\frac{3 t}{t - 3}

\frac{t - 3 - 3 3 t t ^{\frac{2}{3}}}{3 t ^{\frac{2}{3}} ( t - 3 ) ^{2}}

解答ステップ

\frac{d}{d t} (\frac{3 t}{t - 3})

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

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

簡素化

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

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