d/(dt)((s(t))(t(t-6)^2))

解答

\frac{d}{d t} ((s (t)) (t (t - 6)^{2}))

s (4 t^{3} - 36 t^{2} + 72 t)

求解步骤

\frac{d}{d t} ((s (t)) (t (t - 6)^{2}))

化简

(s (t)) (t (t - 6)^{2}) : s t^{2} (t - 6)^{2}

= \frac{d}{d t} (s t^{2} (t - 6)^{2})

将常数提出:

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

= s \frac{d}{d t} (t^{2} (t - 6)^{2})

使用微分乘法定则:

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

f = t^{2}, g = (t - 6)^{2}

= s (\frac{d}{d t} (t^{2}) (t - 6)^{2} + \frac{d}{d t} ((t - 6)^{2}) t^{2})

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

\frac{d}{d t} ((t - 6)^{2}) = 2 (t - 6)

= s (2 t (t - 6)^{2} + 2 (t - 6) t^{2})

乘开

2 t (t - 6)^{2} + 2 (t - 6) t^{2} : 4 t^{3} - 36 t^{2} + 72 t

= s (4 t^{3} - 36 t^{2} + 72 t)

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