What is the d/(dt)(({P}(t)){P}(t)(1-{P}(t))) ?

The d/(dt)(({P}(t)){P}(t)(1-{P}(t))) is 2{P}(t) d/(dt)({P}(t))-3 d/(dt)({P}(t)){P}(t)^2

What is the first d/(dt)(({P}(t)){P}(t)(1-{P}(t))) ?

The first d/(dt)(({P}(t)){P}(t)(1-{P}(t))) is 2{P}(t) d/(dt)({P}(t))-3 d/(dt)({P}(t)){P}(t)^2

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\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}

Solution steps

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

Simplify

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

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

Apply the Product Rule:

(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}

Simplify

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}

\frac{2 x ^{2}}{( 1 - x ) ( 2 - x )}

\frac{1}{3} (π) (h - 8)^{2} h

\frac{8000}{( 1 + x ) ^{3}}

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

(3 x^{2} + 2 x y - y^{2}) d x + (x^{2} - 2 x y - 3 y^{2}) d y

What is the d/(dt)(({P}(t)){P}(t)(1-{P}(t))) ?
The d/(dt)(({P}(t)){P}(t)(1-{P}(t))) is 2{P}(t) d/(dt)({P}(t))-3 d/(dt)({P}(t)){P}(t)^2
What is the first d/(dt)(({P}(t)){P}(t)(1-{P}(t))) ?
The first d/(dt)(({P}(t)){P}(t)(1-{P}(t))) is 2{P}(t) d/(dt)({P}(t))-3 d/(dt)({P}(t)){P}(t)^2