Joshua B. answered • 09/15/21

Economics Ph.D., B.S. in Math. Experienced math tutor.

First, remember that any time you're looking for a **marginal **function in an economics context, you want the **first order derivative** of that function.

Then remember that profit is revenue minus cost: **P(q) = R(q) - C(q)**

Recall the **difference rule**, where the derivative of the difference of two functions equals the difference between those functions' corresponding derivatives:

**P'(q) = R'(q) - C'(q)**

Or to phrase it in a way that econ and business majors can understand, **marginal profit is marginal revenue minus marginal cost:**

**MP(q) = MR(q) - MC(q)**

(Also note that **the difference rule is just a special case of the sum rule**, where the second addend is negative.)

Since the revenue and cost functions are simple polynomials, the **power rule (lower the power of x by 1 and multiply by the previous power), sum rule, and constant rule** should suffice for evaluating the marginal profit function.

I'll write out the power rule for clarity, just as a reminder:

**If f(x) = a*x**^{n}**, where a is constant and n is any nonzero integer, then f'(x) = n*a*x**^{n-1}**.**

Since the degree of the sum (or difference) of the polynomials is the higher degree between them, the profit function should be **quadratic**, since the revenue function is quadratic and the cost function is linear.

**P(q) = a*q**^{2}** + b*q + c** **(where a,b,c are constants)**

That means the marginal profit function is going to be a **linear function.**

**MP(q) = 2a*q + b**

Note again that **the sum/difference rules imply that the order in which you subtract and differentiate is irrelevant.** So you could just as easily differentiate the revenue and cost functions separately, and then subtract the derivatives to get the marginal profit function.