Chapter 3.4, Problem 39E

To determine

To explain:The derivation of marginal profit in terms of marginal revenue and marginal cost.

Answer to Problem 39E

Formula of marginal profit is $mariginalprofit=marginalrevenue-marginalcost$ .

Explanation of Solution

Given information:

Need to use sum and difference rule to derive formula for marginal profit.

Assume revenue generated by selling $x$ items be represented by function $r\left(x\right)$ .

So marginal revenue can be represented by $\frac{d}{dx}r\left(x\right)$

Assume that cost of $x$ items be represented by function $c\left(x\right)$ .

So marginal costcan be represented by $\frac{d}{dx}c\left(x\right)$

Assume profit of selling $x$ items be represented by function $P\left(x\right)$ .

As $\text{Profit=revenue}\text{-}\text{cost}$

  $P\left(x\right)=r\left(x\right)-c\left(x\right)$

Evaluate marginal profit that is $\frac{d}{dx}P\left(x\right)$ by differentiating both sides of equation.

  $\frac{d}{dx}P\left(x\right)=\frac{d}{dx}\left(r\left(x\right)-c\left(x\right)\right)$

Apply sum and difference rule

  $\frac{d}{dx}P\left(x\right)=\frac{d}{dx}r\left(x\right)-\frac{d}{dx}c\left(x\right)$

  $mariginalprofit=marginalrevenue-marginalcost$

Hence required derived formula of marginal profit using sum and difference rule is $\underset{\_}{mariginalprofit=marginalrevenue-marginalcost}$ .