## Example

Zoe's lemonade stand satisfies a demand characterized by:

$$ P = 10 - 2 Q $$Her revenue is

$$ R \left( Q \right) = PQ = \left( 10 - 2 Q \right) Q = 10 Q - 2 Q^2 $$The marginal revenue is the derivative of the revenue `R \left( Q \right)`:

$$ MR \left( Q \right) = \frac{d R \left( Q \right)}{d Q} = 10 - 2 \times 2 Q = 10 - 4 Q $$### Question

The inverse demand for lemonade is `P = 74 - 4Q`.

What is Zoe's marginal revenue?

First, her revenue is

$$ R \left( Q \right) = PQ = (74 - 4Q) Q = 74 Q - 4Q^2 $$So her marginal revenue is

$$ MR \left( Q \right) = \frac{d R \left( Q \right)}{d Q} = 74 - 4 \times 2 Q = 74 - 8Q $$