derivative h(t)=(t^4-1)^3(t^3+1)^4

解

導関数

h (t) = (t^{4} - 1)^{3} (t^{3} + 1)^{4}

12 t^{3} (t^{4} - 1)^{2} (t^{3} + 1)^{4} + 12 t^{2} (t^{3} + 1)^{3} (t^{4} - 1)^{3}

解答ステップ

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

積の法則を適用:

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

f = (t^{4} - 1)^{3}, g = (t^{3} + 1)^{4}

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

\frac{d}{d t} ((t^{4} - 1)^{3}) = 12 t^{3} (t^{4} - 1)^{2}

\frac{d}{d t} ((t^{3} + 1)^{4}) = 12 t^{2} (t^{3} + 1)^{3}

= 12 t^{3} (t^{4} - 1)^{2} (t^{3} + 1)^{4} + 12 t^{2} (t^{3} + 1)^{3} (t^{4} - 1)^{3}