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To find the **marginal profit function** \( P'(x) \), we need to understand that the **profit function** \( P(x) \) is defined as:

\[
P(x) = R(x) - C(x)
\]

where:
- \( R(x) \) is the revenue function, and
- \( C(x) \) is the cost function.

From the problem, we have:
- \( C(x) = 221 + 0.6x \)
- \( R(x) = 2x - 0.09x^2 \)

Now, the **marginal profit** is the derivative of the profit function:

\[
P'(x) = R'(x) - C'(x)
\]

### Step 1: Find the derivative of \( R(x) \) (Marginal Revenue)
\[
R(x) = 2x - 0.09x^2
\]
Taking the derivative with respect to \( x \):
\[
R'(x) = 2 - 0.18x
\]

### Step 2: Find the derivative of \( C(x) \) (Marginal Cost)
\[
C(x) = 221 + 0.6x
\]
Taking the derivative with respect to \( x \):
\[
C'(x) = 0.6
\]

### Step 3: Compute \( P'(x) \) (Marginal Profit)
Now, subtract the marginal cost from the marginal revenue:
\[
P'(x) = R'(x) - C'(x) = (2 - 0.18x) - 0.6
\]
Simplifying:
\[
P'(x) = 1.4 - 0.18x
\]

So, the marginal profit function is:
\[
P'(x) = 1.4 - 0.18x
\]

Would you like a more detailed explanation on any part of the solution? Here are some related questions for further practice:

1. How would the profit function \( P(x) \) be computed from \( R(x) \) and \( C(x) \)?
2. What is the significance of the marginal cost in business decision-making?
3. How does the shape of the revenue function \( R(x) \) affect the marginal profit?
4. How would you calculate the exact profit at a specific \( x \) value?
5. What would happen to the marginal profit if the cost function changed to \( C(x) = 300 + 0.4x \)?

**Tip:** The marginal profit is a powerful tool in economics, helping businesses decide optimal production levels by indicating how much profit increases or decreases when production is changed.

Find the marginal profit function if cost and revenue are given by C(x) = 221 + 0.6x and R(x) = 2x - 0.09x^2.

This solution explains how to find the marginal profit function from given cost and revenue functions using derivatives. The problem involves calculating the marginal revenue and marginal cost and subtracting them to find the marginal profit. Suitable for high school calculus students or those in introductory economics courses.

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