d/(dt)(({P}(t,M))(r(t,M){P}(t,M)(M-{P}(t,M))))

解答

\frac{d}{d t} ((P (t, M)) (r (t, M) P (t, M) (M - P (t, M))))

\frac{d}{d t} (P (t, M)) r (t, M) P (t, M) (M - P (t, M)) + (\frac{d}{d t} (r (t, M)) P (t, M) (M - P (t, M)) + (\frac{d}{d t} (P (t, M)) (M - P (t, M)) + (0 - \frac{d}{d t} (P (t, M))) P (t, M)) r (t, M)) P (t, M)

求解步骤

\frac{d}{d t} ((P (t, M)) (r (t, M) P (t, M) (M - P (t, M))))

将

M

视为常数

使用微分乘法定则:

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

f = P (t, M), g = r (t, M) P (t, M) (M - P (t, M))

= \frac{d}{d t} (P (t, M)) r (t, M) P (t, M) (M - P (t, M)) + \frac{d}{d t} (r (t, M) P (t, M) (M - P (t, M))) P (t, M)

\frac{d}{d t} (r (t, M) P (t, M) (M - P (t, M))) = \frac{d}{d t} (r (t, M)) P (t, M) (M - P (t, M)) + (\frac{d}{d t} (P (t, M)) (M - P (t, M)) + (0 - \frac{d}{d t} (P (t, M))) P (t, M)) r (t, M)

= \frac{d}{d t} (P (t, M)) r (t, M) P (t, M) (M - P (t, M)) + (\frac{d}{d t} (r (t, M)) P (t, M) (M - P (t, M)) + (\frac{d}{d t} (P (t, M)) (M - P (t, M)) + (0 - \frac{d}{d t} (P (t, M))) P (t, M)) r (t, M)) P (t, M)

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