d/(dt)((x)(kx(t)(N-x)))

解

\frac{d}{d t} ((x (t)) (k x (t) (N - x (t))))

k (2 N x (t) \frac{d}{d t} (x (t)) - 3 \frac{d}{d t} (x (t)) x (t)^{2})

解答ステップ

\frac{d}{d t} ((x (t)) (k x (t) (N - x (t))))

簡素化

(x (t)) (k x (t) (N - x (t))) : k x (t)^{2} (N - x (t))

= \frac{d}{d t} (k x (t)^{2} (N - x (t)))

定数を除去:

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

= k \frac{d}{d t} (x (t)^{2} (N - x (t)))

積の法則を適用:

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

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

= k (\frac{d}{d t} (x (t)^{2}) (N - x (t)) + \frac{d}{d t} (N - x (t)) x (t)^{2})

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

\frac{d}{d t} (N - x (t)) = - \frac{d}{d t} (x (t))

= k (2 x (t) \frac{d}{d t} (x (t)) (N - x (t)) + (- \frac{d}{d t} (x (t))) x (t)^{2})

簡素化

k (2 x (t) \frac{d}{d t} (x (t)) (N - x (t)) + (- \frac{d}{d t} (x (t))) x (t)^{2}) : k (2 N x (t) \frac{d}{d t} (x (t)) - 3 \frac{d}{d t} (x (t)) x (t)^{2})

= k (2 N x (t) \frac{d}{d t} (x (t)) - 3 \frac{d}{d t} (x (t)) x (t)^{2})