d/(dt)(({P}(t))(k{P}(t)(1200-{P}(t))))

解答

\frac{d}{d t} ((P (t)) (k P (t) (1200 - P (t))))

k (2400 P (t) \frac{d}{d t} (P (t)) - 3 \frac{d}{d t} (P (t)) P (t)^{2})

求解步骤

\frac{d}{d t} ((P (t)) (k P (t) (1200 - P (t))))

化简

(P (t)) (k P (t) (1200 - P (t))) : k P (t)^{2} (1200 - P (t))

= \frac{d}{d t} (k P (t)^{2} (1200 - P (t)))

将常数提出:

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

= k \frac{d}{d t} (P (t)^{2} (1200 - P (t)))

使用微分乘法定则:

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

f = P (t)^{2}, g = 1200 - P (t)

= k (\frac{d}{d t} (P (t)^{2}) (1200 - P (t)) + \frac{d}{d t} (1200 - P (t)) P (t)^{2})

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

\frac{d}{d t} (1200 - P (t)) = - \frac{d}{d t} (P (t))

= k (2 P (t) \frac{d}{d t} (P (t)) (1200 - P (t)) + (- \frac{d}{d t} (P (t))) P (t)^{2})

化简

k (2 P (t) \frac{d}{d t} (P (t)) (1200 - P (t)) + (- \frac{d}{d t} (P (t))) P (t)^{2}) : k (2400 P (t) \frac{d}{d t} (P (t)) - 3 \frac{d}{d t} (P (t)) P (t)^{2})

= k (2400 P (t) \frac{d}{d t} (P (t)) - 3 \frac{d}{d t} (P (t)) P (t)^{2})

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