# Finding the derivative of a revenue function

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1. Oct 30, 2014

### tech_chic

1. The problem statement, all variables and given/known data
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?

2. Relevant equations
r=pq

3. 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.

2. Oct 30, 2014

### SteamKing

Staff Emeritus
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?

3. Oct 30, 2014

### tech_chic

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

4. Oct 30, 2014

### SteamKing

Staff Emeritus
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?

5. Oct 30, 2014

### 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.

6. Oct 30, 2014

### 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

7. Oct 30, 2014

### SteamKing

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

8. Oct 30, 2014

### Ray Vickson

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.