What steps do I take to solve the answer (Answer is in thousands)

Question

A company has determined the demand curve for their product is q=2000−p^2, where p is the price in dollars, and q is the quantity in thousands. If the price is decreasing by $0.25 per week, find the rate at which demand is changing when the price is $10.

Answer 1

q=2000-p^2

q'=-2p=-2(10)=-20 = rate of change in quantity demanded

the following are just extra calculations

p^2=2000-q

p= sqr(2000-q)

p' = -1/sqr(2000-q)= -.25

-4=sqr(2000-q)

2000-q=16

q=1984

p=sqr(2000-1984)= sqr16= 4

p=10

10^2=2000-q

q=1900

p'=-1/(2000-1900)=-1/100= -.01

q'=-2p=-2(4)=-8

as p increases from 4 to 10

q decreases from 1984 to 1900

-84 per $6 increase -84/6 = -14

average change = (-20-8)/2 =-14

at p=4, q'=-8

at p=7 q'=-14

at p=10 q'=-20

but all you need is

q=2000-p^2

q'=-2p=-2(10)=-20

= rate of change in quantity demanded when price = $10