Question

A company can charge a price of 89 dollars per unit when they sell 520 units. If the price increases by 2 dollars then the number of units that they sell drops by 5 . The revenue function for this company is as follows: R(x)=(89+2x)(520−5x) where x is the number of additional units sold beyond 520 . Use the Product Rule to find the derivative of R(x) R' (x)=1

Answer 1

The derivative of the revenue function R(x) = (89 + 2x)(520 - 5x) is R'(x) = 595 - 20x, found using the Product Rule. This represents the marginal revenue which shows additional revenue from selling one more unit.

Step-by-step explanation:

A student has posed a question about finding the derivative of a revenue function using the Product Rule. Specifically, they have provided the revenue function R(x) = (89 + 2x)(520 - 5x) where x represents the number of additional units sold beyond 520.

To find the derivative of the revenue function R'(x) using the Product Rule, we apply the rule which states that the derivative of a product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function.

Applying this rule to our function, we get:

R'(x) = (89 + 2x)(-5) + (520 - 5x)(2)

That simplifies to:

R'(x) = -445 - 10x + 1040 - 10x

R'(x) = 595 - 20x

This derivative R'(x) represents the marginal revenue, which is the additional revenue generated from selling one more unit beyond the initial 520 units.