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Let's break down and solve each part of the problem step by step.

### Given:
- Revenue: \( R(x) = 4e^{2x} \)
- Cost: \( C(x) = \ln(5x + 3) \)

We are asked to find:
a. The marginal revenue  
b. The marginal cost  
c. Profit  
d. Marginal profit

### a. Marginal Revenue:
The marginal revenue is the derivative of the revenue function \( R(x) \).  
\[
R'(x) = \frac{d}{dx} \left( 4e^{2x} \right)
\]
Using the chain rule:
\[
R'(x) = 4 \cdot 2e^{2x} = 8e^{2x}
\]
Thus, the **marginal revenue** is:
\[
R'(x) = 8e^{2x}
\]

### b. Marginal Cost:
The marginal cost is the derivative of the cost function \( C(x) \).
\[
C'(x) = \frac{d}{dx} \left( \ln(5x + 3) \right)
\]
Using the chain rule:
\[
C'(x) = \frac{1}{5x + 3} \cdot 5 = \frac{5}{5x + 3}
\]
Thus, the **marginal cost** is:
\[
C'(x) = \frac{5}{5x + 3}
\]

### c. Profit:
The profit \( P(x) \) is the difference between revenue and cost:
\[
P(x) = R(x) - C(x)
\]
Substitute the given functions:
\[
P(x) = 4e^{2x} - \ln(5x + 3)
\]
Thus, the **profit** function is:
\[
P(x) = 4e^{2x} - \ln(5x + 3)
\]

### d. Marginal Profit:
The marginal profit is the derivative of the profit function:
\[
P'(x) = R'(x) - C'(x)
\]
From previous results:
\[
P'(x) = 8e^{2x} - \frac{5}{5x + 3}
\]
Thus, the **marginal profit** is:
\[
P'(x) = 8e^{2x} - \frac{5}{5x + 3}
\]

---

Would you like further clarifications on any of these steps or additional details on specific parts? Here are some related questions:

1. How does the exponential term in revenue affect the overall profit as \( x \) increases?
2. What happens to marginal revenue and marginal cost as \( x \) approaches infinity?
3. How would profit change if the cost function were quadratic instead of logarithmic?
4. Can you explain the significance of marginal profit in terms of decision-making?
5. How do chain rule and product rule apply in more complex revenue and cost functions?

Tip: Remember that marginal values (revenue, cost, profit) help you understand the rate of change of these quantities with respect to production or sales quantity.

If Revenue is given by the equation R(x) = 4e^2x and Cost is given by the equation C(x) = ln(5x+3), find: a) The marginal revenue b) The marginal cost c) Profit d) Marginal profit

Learn how to calculate marginal revenue, marginal cost, profit, and marginal profit when given exponential and logarithmic revenue and cost functions. This problem involves key calculus concepts such as differentiation using the chain rule. Suitable for introductory undergraduate calculus courses.

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