Find the corresponding rate of decrease in demand for...?

Question

At a price of $p the demand x per month (in multiples of 100) for a new piece of software is given by x^2 + 2xp + 7p^2 = 1000.

Because of its popularity, the manufacturer is increasing the price at a rate of 20 cents per month. Find the corresponding rate of decrease in demand for the software when the software costs $10.

I am pretty sure the x variable is 10, like p. I used the quadratic formula to get this. I think i'm confusing some step in the middle where you need to use the product rule.

Answer 1

Your equation has the form:

Ax^2 + Bxp + Cp^2 = D

This is a quadratic so you can use the quadratic formula to get x as a function p. (This function involves square roots, etc.)

Then you need to use the chain rule. If H(t) = F(G(t)) then
dH/dt = (dF/dU)(dG/dt)

In this case G(t) corresponds to p(t). You know the value of G(t) and the value of G'(t).

F(U) corresponds to x as a function of p.

So compute dx/dp; multiply it by dp/dt; and you are done.

As for dx/dp, just remember the basic rules:

d(f+g)/dx = df/dx + dg/dx
dx^r/dx = rx^(r-1) (for all r, not just integers)
d(f(x)g(x))/dx = f'(x)g(x) + f(x)g'(x)

Answer 2

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