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### Problem Description:

The problem is about determining the rate of change of the profit when increasing production. Here's a translation and breakdown of the problem:

**Translation:**

A chemical products manufacturer notes that the cost per week of producing \( x \) tons of a certain fertilizer is given by:
\[ C(x) = 20,000 + 40x \text{ (dollars)} \]
and the revenue obtained from selling \( x \) tons is given by:
\[ R(x) = 100x + 0.01x^2 \text{ (dollars)} \]

The company currently produces \( 3,000 \) tons per week but is considering increasing production by \( 500 \) tons per week. Determine the rate of change in profit per unit of the additional production.

### Solution:

The profit \( P(x) \) is given by the difference between revenue \( R(x) \) and cost \( C(x) \):
\[ P(x) = R(x) - C(x) \]

Substituting the given functions:
\[ P(x) = (100x + 0.01x^2) - (20,000 + 40x) \]

Simplifying the expression:
\[ P(x) = 100x + 0.01x^2 - 20,000 - 40x \]
\[ P(x) = 60x + 0.01x^2 - 20,000 \]

To find the rate of change of the profit with respect to the increase in production, we need to compute the derivative of \( P(x) \) with respect to \( x \):
\[ \frac{dP}{dx} = \frac{d}{dx} \left( 60x + 0.01x^2 - 20,000 \right) \]

Differentiating:
\[ \frac{dP}{dx} = 60 + 0.02x \]

Now, substitute \( x = 3000 \) to find the rate of change of the profit when the current production is \( 3000 \) tons:
\[ \frac{dP}{dx} \bigg|_{x=3000} = 60 + 0.02(3000) \]
\[ \frac{dP}{dx} \bigg|_{x=3000} = 60 + 60 = 120 \text{ dollars per ton} \]

### Conclusion:

The rate of change of the profit per unit increase in production at \( 3000 \) tons is \( 120 \) dollars per ton. Therefore, increasing the production by 500 tons per week would increase the profit at a rate of \( 120 \times 500 = 60,000 \) dollars per week.

### Do you want details or have any specific questions?

### Relative Questions:
1. How does the cost function \( C(x) \) impact the overall profit as \( x \) increases?
2. What would be the impact on profit if the revenue function \( R(x) \) had a different quadratic term?
3. What is the significance of the derivative \( \frac{dP}{dx} \) in real-world business decisions?
4. How would the profit change if production is decreased instead of increased?
5. Can you explain how the second derivative of \( P(x) \) relates to the concavity of the profit function?

### Tip:
When analyzing profit functions, the first derivative gives you the rate of change of profit with respect to production, while the second derivative provides insights into whether this rate of change is increasing or decreasing.

Understand how to calculate the rate of change of profit when increasing production in business scenarios. Explore the application of calculus in determining profit functions, revenue, and cost relationships. Suitable for undergraduate level studies.

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