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

Lösung

\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}

Schritte zur Lösung

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

Vereinfache

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

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

Wende die Produktregel an:

(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}

Vereinfache

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}

e x p an d \frac{2 x ^{2}}{( 1 - x ) ( 2 - x )}

e x p an d \frac{1}{3} (π) (h - 8)^{2} h

e x p an d \frac{8000}{( 1 + x ) ^{3}}

\frac{d ^{2}}{d x ^{2}} ((y) (cos (5 x) - 4))

e x p an d \frac{507}{s ( s ^{2} - 4 s + 13 ) ^{2}}