implicit differentiation?

Question

the supply equation for a certain commodity is
x^2 = 10^6p(3p+20), where x units are supplied per month when p in dollars is the price per unit. find the rate of change of supply with respect to the price when the price is $20 and 1 million units are supplied

Answer 1

Call the function x(p).

If we want dx/dp, take the derivative wrt p of each term:

(2x)dx/dp = (10^6)(3p + 20) + (10^6p)3

or

dx/dp = 10^6(6p + 20)/(2x)

Now put p = 20 and x = 1000000 into the above eqn.

dx/dp = 10^6(140)/2000000

=70 [edited slightly for an algebra problem. ;) ]

Hope that's all clear.

Answer 2

2x*dx/dp=10^6(3p+20+3p) so
dx/dp= 10^6(6p+20)/2x = 10^6(140)/2*10^6 = 70 units/dollar