A price $p$ (in dollars) and demand $x$ for a product are related by $$\left(2x^2\right)-2xp+50p^2 = 20600$$ If the price is increasing at a rate of $2$ dollars per month when the price is $20$ dollars, find the rate of change of the demand.
I was a little confused on how to proceed with this question. Am I supposed to use implicit differentiation (with the $x$ serving the same purpose as a $y$) and then find the derivative of $x$?
This is the implicit differentiation I tried:
$$4x\frac{dx}{dp}-2\frac{dx}{dp}+100p = 0$$
$$4x\frac{dx}{dp}-2\frac{dx}{dp} = -100p$$
$$\frac{dx}{dp}(4x-2) = -100p$$
$$\frac{dx}{dp} = \frac{-100p}{4x-2}$$
I believe this is the derivative I am looking for (thought not entirely sure) but I am not sure what values of $p$ and $x$ to input, as I am supposed to get a numerical final answer.
Any help?
Best Answer
Guide:
$$2x^2-2xp+50p^2=20600$$
Differentiate with respect to $t$.
We have information about $x, p, \frac{dp}{dt}$, and you are interested in finding $\frac{dx}{dt}$.
Note that when $p=20$, you can compute the corresponding $x$.