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

解

\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)

解答ステップ

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

M

を定数として扱う

積の法則を適用:

(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)