derivative y=2t(3t^2-5)^3

解

導関数

y = 2 t (3 t^{2} - 5)^{3}

2 ((3 t^{2} - 5)^{3} + 18 t^{2} (3 t^{2} - 5)^{2})

解答ステップ

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

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

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

積の法則を適用:

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

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

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

\frac{d t}{d t} = 1

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

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

簡素化

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

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