According to economic theory, the demand x for a quantity in a free market decreases as the price p increases. Suppose that the number x of DVD players people are willing to buy per week from a retail chain at a price of $p is given by

x$=4000,10<=$p$<=70.0\mathrm{.}1$p$+1$

$($a$)$ Find dx$/$dp$.$

$($b$)$ Find the demand and the instantaneous rate of change of demand with respect to price when the

price is $$40.$ Write a brief interpretation of these results.

$($c$)$ Use the results from part $($b$)$ to estimate the demand if the price is increased to $$41.$