I'm brushing up on calculus and started the differential equations section and found this problem and am very confused about the answer justification (not about why its (B)). Why does the answer not use implicit differentiation? Why is it not
$$\frac{d^2P}{dt^2}=2P'-0.004PP'?$$
I can get the same answer for $y''=0$ by factoring out $P'$ and dividing out $P'$, but it still doesn't answer why they differentiated like they did.
Also, they say their max population is $P=500$, but shouldn't max population be the limiting value of $P=1000$, with their $P=500$ instead being where it's the maximum growth rate?
From Barron's AP Calculus Premium 15th Edition