d/(dt)((a(t))(a(t)^4+t^4-6a(t)^2t))

解

\frac{d}{d t} ((a (t)) (a (t)^{4} + t^{4} - 6 a (t)^{2} t))

a (5 a t^{4} + 5 t^{4} - 24 a t^{3})

解答ステップ

\frac{d}{d t} ((a (t)) (a (t)^{4} + t^{4} - 6 a (t)^{2} t))

簡素化

(a (t)) (a (t)^{4} + t^{4} - 6 a (t)^{2} t) : a t (a t^{4} + t^{4} - 6 a t^{3})

= \frac{d}{d t} (a t (a t^{4} + t^{4} - 6 a t^{3}))

定数を除去:

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

= a \frac{d}{d t} (t (a t^{4} + t^{4} - 6 a t^{3}))

積の法則を適用:

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

f = t, g = a t^{4} + t^{4} - 6 a t^{3}

= a (\frac{d t}{d t} (a t^{4} + t^{4} - 6 a t^{3}) + \frac{d}{d t} (a t^{4} + t^{4} - 6 a t^{3}) t)

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

\frac{d}{d t} (a t^{4} + t^{4} - 6 a t^{3}) = 4 a t^{3} + 4 t^{3} - 18 a t^{2}

= a (1 \cdot (a t^{4} + t^{4} - 6 a t^{3}) + (4 a t^{3} + 4 t^{3} - 18 a t^{2}) t)

簡素化

a (1 \cdot (a t^{4} + t^{4} - 6 a t^{3}) + (4 a t^{3} + 4 t^{3} - 18 a t^{2}) t) : a (5 a t^{4} + 5 t^{4} - 24 a t^{3})

= a (5 a t^{4} + 5 t^{4} - 24 a t^{3})