Let's further simplify the derivatives obtained in the previous solution. 

### First Derivative:

Given:
\[ f(x) = e^{2x} + 3x \cos(x) \]

The first derivative is:
\[ f'(x) = 2e^{2x} - 3x \sin(x) + 3 \cos(x) \]

This expression is already simplified, but we can rewrite it as:
\[ f'(x) = 2e^{2x} + 3 \cos(x) - 3x \sin(x) \]

### Second Derivative:

Given the first derivative:
\[ f'(x) = 2e^{2x} - 3x \sin(x) + 3 \cos(x) \]

Now, differentiate this again to find the second derivative.

1. Differentiate \( 2e^{2x} \):
\[ \frac{d}{dx} \left( 2e^{2x} \right) = 4e^{2x} \]

2. Differentiate \( -3x \sin(x) \) using the product rule:
\[ \frac{d}{dx} \left( -3x \sin(x) \right) = -3 \left( x \frac{d}{dx} \left( \sin(x) \right) + \sin(x) \frac{d}{dx} \left( x \right) \right) = -3 \left( x \cos(x) + \sin(x) \right) \]

3. Differentiate \( 3 \cos(x) \):
\[ \frac{d}{dx} \left( 3 \cos(x) \right) = 3 \frac{d}{dx} \left( \cos(x) \right) = 3 (-\sin(x)) = -3 \sin(x) \]

Combining these results, we get:
\[ f''(x) = 4e^{2x} - 3 \left( x \cos(x) + \sin(x) \right) - 3 \sin(x) \]
\[ f''(x) = 4e^{2x} - 3x \cos(x) - 3 \sin(x) - 3 \sin(x) \]
\[ f''(x) = 4e^{2x} - 3x \cos(x) - 6 \sin(x) \]

This expression is already in its simplest form.

So, the simplified forms are:

**First Derivative:**
\[ f'(x) = 2e^{2x} + 3 \cos(x) - 3x \sin(x) \]

**Second Derivative:**
\[ f''(x) = 4e^{2x} - 3x \cos(x) - 6 \sin(x) \]

Would you like further details or have any questions about these solutions?

### Related Questions:

1. How do you differentiate \( e^{kx} \) where \( k \) is a constant?
2. What are the steps to differentiate a product of two functions?
3. How do you differentiate trigonometric functions like \( \cos(x) \) and \( \sin(x) \)?
4. How can you verify the correctness of a derivative?
5. What are some applications of the second derivative in real-world problems?
6. How do the derivatives of exponential functions behave as \( x \) approaches infinity?
7. Can you find the critical points using the first derivative?
8. What information can the second derivative provide about the function's graph?

### Tip:
When simplifying derivatives, always combine like terms and ensure that all parts of the expression are fully expanded before combining. This helps in reducing the expression to its simplest form.

Learn how to simplify the first and second derivatives of f(x) = e^(2x) + 3x * cos(x). Understand the rules for differentiating exponential and trigonometric functions. Suitable for advanced mathematics students.

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