What is the d/(dt)((x)(kx(t)(1000-x))) ?

The d/(dt)((x)(kx(t)(1000-x))) is k(2000{x}(t) d/(dt)({x}(t))-3 d/(dt)({x}(t)){x}(t)^2)

What is the first d/(dt)((x)(kx(t)(1000-x))) ?

The first d/(dt)((x)(kx(t)(1000-x))) is k(2000{x}(t) d/(dt)({x}(t))-3 d/(dt)({x}(t)){x}(t)^2)

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\frac{d}{d t} ((x (t)) (k x (t) (1000 - x (t))))

k (2000 x (t) \frac{d}{d t} (x (t)) - 3 \frac{d}{d t} (x (t)) x (t)^{2})

Solution steps

\frac{d}{d t} ((x (t)) (k x (t) (1000 - x (t))))

Simplify

(x (t)) (k x (t) (1000 - x (t))) : k x (t)^{2} (1000 - x (t))

= \frac{d}{d t} (k x (t)^{2} (1000 - x (t)))

Take the constant out:

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

= k \frac{d}{d t} (x (t)^{2} (1000 - x (t)))

Apply the Product Rule:

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

f = x (t)^{2}, g = 1000 - x (t)

= k (\frac{d}{d t} (x (t)^{2}) (1000 - x (t)) + \frac{d}{d t} (1000 - 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} (1000 - x (t)) = - \frac{d}{d t} (x (t))

= k (2 x (t) \frac{d}{d t} (x (t)) (1000 - x (t)) + (- \frac{d}{d t} (x (t))) x (t)^{2})

Simplify

k (2 x (t) \frac{d}{d t} (x (t)) (1000 - x (t)) + (- \frac{d}{d t} (x (t))) x (t)^{2}) : k (2000 x (t) \frac{d}{d t} (x (t)) - 3 \frac{d}{d t} (x (t)) x (t)^{2})

= k (2000 x (t) \frac{d}{d t} (x (t)) - 3 \frac{d}{d t} (x (t)) x (t)^{2})

\frac{5 x - 2}{( x - 2 ) ( x + 2 )}

\frac{2}{y ( y + 8 )}

e^{(3 t - 3)} h (t - 1)

- \frac{1}{4 ( 2 x - 1 )}

a (- 4, 2) b (8, - 3)

What is the d/(dt)((x)(kx(t)(1000-x))) ?
The d/(dt)((x)(kx(t)(1000-x))) is k(2000{x}(t) d/(dt)({x}(t))-3 d/(dt)({x}(t)){x}(t)^2)
What is the first d/(dt)((x)(kx(t)(1000-x))) ?
The first d/(dt)((x)(kx(t)(1000-x))) is k(2000{x}(t) d/(dt)({x}(t))-3 d/(dt)({x}(t)){x}(t)^2)