derivative (t+5)^{2/3}(3t^2-2)^3

解

導関数

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

\frac{2 ( 3 t ^{2} - 2 ) ^{3} + 54 t ( 3 t ^{2} - 2 ) ^{2} ( t + 5 )}{3 ( t + 5 ) ^{\frac{1}{3}}}

解答ステップ

\frac{d}{d t} ((t + 5)^{\frac{2}{3}} (3 t^{2} - 2)^{3})

積の法則を適用:

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

f = (t + 5)^{\frac{2}{3}}, g = (3 t^{2} - 2)^{3}

= \frac{d}{d t} ((t + 5)^{\frac{2}{3}}) (3 t^{2} - 2)^{3} + \frac{d}{d t} ((3 t^{2} - 2)^{3}) (t + 5)^{\frac{2}{3}}

\frac{d}{d t} ((t + 5)^{\frac{2}{3}}) = \frac{2}{3 ( t + 5 ) ^{\frac{1}{3}}}

\frac{d}{d t} ((3 t^{2} - 2)^{3}) = 18 t (3 t^{2} - 2)^{2}

= \frac{2}{3 ( t + 5 ) ^{\frac{1}{3}}} (3 t^{2} - 2)^{3} + 18 t (3 t^{2} - 2)^{2} (t + 5)^{\frac{2}{3}}

簡素化

\frac{2}{3 ( t + 5 ) ^{\frac{1}{3}}} (3 t^{2} - 2)^{3} + 18 t (3 t^{2} - 2)^{2} (t + 5)^{\frac{2}{3}} : \frac{2 ( 3 t ^{2} - 2 ) ^{3} + 54 t ( 3 t ^{2} - 2 ) ^{2} ( t + 5 )}{3 ( t + 5 ) ^{\frac{1}{3}}}

= \frac{2 ( 3 t ^{2} - 2 ) ^{3} + 54 t ( 3 t ^{2} - 2 ) ^{2} ( t + 5 )}{3 ( t + 5 ) ^{\frac{1}{3}}}