d/(dt)(({P}(t)){P}(t)(1-{P}(t)))

解

\frac{d}{d t} ((P (t)) P (t) (1 - P (t)))

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

解答ステップ

\frac{d}{d t} ((P (t)) P (t) (1 - P (t)))

簡素化

(P (t)) P (t) (1 - P (t)) : P (t)^{2} (1 - P (t))

= \frac{d}{d t} (P (t)^{2} (1 - P (t)))

積の法則を適用:

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

f = P (t)^{2}, g = 1 - P (t)

= \frac{d}{d t} (P (t)^{2}) (1 - P (t)) + \frac{d}{d t} (1 - 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} (1 - P (t)) = - \frac{d}{d t} (P (t))

= 2 P (t) \frac{d}{d t} (P (t)) (1 - P (t)) + (- \frac{d}{d t} (P (t))) P (t)^{2}

簡素化

2 P (t) \frac{d}{d t} (P (t)) (1 - P (t)) + (- \frac{d}{d t} (P (t))) P (t)^{2} : 2 P (t) \frac{d}{d t} (P (t)) - 3 \frac{d}{d t} (P (t)) P (t)^{2}

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