Finding the derivative of a revenue function

[tech_chic]

Homework Statement

The revenue function for a product is r = 8x where r is in dollars and x is the number of units sold. the demand function is q = -1/4p + 10000 where q units can be sold when selling price is p. what is dr/dp?

Homework Equations

r=pq

The Attempt at a Solution

I substituted the values into r=pq.

8x= -1/4p^2 + 10000p

i'm not sure which rule to use from here to find the derivative or if i overcomplicated the question.

 

[SteamKing]

Homework Statement

The revenue function for a product is r = 8x where r is in dollars and x is the number of units sold. the demand function is q = -1/4p + 10000 where q units can be sold when selling price is p. what is dr/dp?

Homework Equations

r=pq

The Attempt at a Solution

I substituted the values into r=pq.

8x= -1/4p^2 + 10000p

i'm not sure which rule to use from here to find the derivative or if i overcomplicated the question.

You got slightly off track.

r = pq and q is a function of p. In essence, you have r = p * f(p). How would you find dr/dp now?

 

[tech_chic]

I'm thinking I would use the product rule. but I'm still confused on the equation 'r = p * f (p).

 

[SteamKing]

I'm thinking I would use the product rule. but I'm still confused on the equation 'r = p * f (p).

It's not clear why you are still confused.

The product rule is OK, but is that rule all that you need to find dr/dp?

 

[tech_chic]

oh yeah the chain rule as well. and I'm still partially confused because I'm not sure how to use the rules on the equation.

 

[BvU]

Can someone explain why "r = 8x where r is in dollars and x is the number of units sold" and "there are 10000 - p/4 units sold if the price is p" does not lead to r = 8p ?

And hence dr/dp = -2 and revenue is maximum ($ 80000) if you give the product away for free ?

It doesn't sound logical, so either I am misunderstanding, or I am misinformed in the problem formulation

 

[SteamKing]

oh yeah the chain rule as well. and I'm still partially confused because I'm not sure how to use the rules on the equation.

Start with applying the chain rule to the equation r = p * f(p). Don't worry about substituting for f(p) at first.

 

[Ray Vickson]

Homework Statement

The revenue function for a product is r = 8x where r is in dollars and x is the number of units sold. the demand function is q = -1/4p + 10000 where q units can be sold when selling price is p. what is dr/dp?

Homework Equations

r=pq

The Attempt at a Solution

I substituted the values into r=pq.

8x= -1/4p^2 + 10000p

i'm not sure which rule to use from here to find the derivative or if i overcomplicated the question.

There is something seriously wrong with your problem statement. If, in fact, the revenue really is r = 8x (x = number sold), then the price must = 8 (=the coefficient of the number sold in the expression for r), and you have x = 10000(8) - (1/4)8^2 = 79984 and r = 8x = 639872. Nothing is varying, so the derivative = 0.

On the other hand, if you want the derivative of r = p*q(p) at p = 8, that is an entirely different question. The equation you wrote (8x = -1/4 p^2 + 10000 p) makes no sense.