The given function is:

\[
y(t) = C_1 e^{-3t} \cos(4t) + C_2 e^{-3t} \sin(4t)
\]

To find the derivative \( y'(t) \), we'll use the product rule, along with the derivatives of \( e^{-3t} \), \( \cos(4t) \), and \( \sin(4t) \).

### Step-by-step:

1. The derivative of \( e^{-3t} \) is:
   \[
   \frac{d}{dt} e^{-3t} = -3 e^{-3t}
   \]

2. The derivative of \( \cos(4t) \) is:
   \[
   \frac{d}{dt} \cos(4t) = -4 \sin(4t)
   \]

3. The derivative of \( \sin(4t) \) is:
   \[
   \frac{d}{dt} \sin(4t) = 4 \cos(4t)
   \]

Now, apply the product rule to each term:

\[
\frac{d}{dt} \left( C_1 e^{-3t} \cos(4t) \right) = C_1 \left( -3 e^{-3t} \cos(4t) + e^{-3t} (-4 \sin(4t)) \right)
\]

\[
\frac{d}{dt} \left( C_2 e^{-3t} \sin(4t) \right) = C_2 \left( -3 e^{-3t} \sin(4t) + e^{-3t} (4 \cos(4t)) \right)
\]

Now combine both terms:

\[
y'(t) = C_1 \left( -3 e^{-3t} \cos(4t) - 4 e^{-3t} \sin(4t) \right) + C_2 \left( -3 e^{-3t} \sin(4t) + 4 e^{-3t} \cos(4t) \right)
\]

Factor out \( e^{-3t} \):

\[
y'(t) = e^{-3t} \left[ C_1 (-3 \cos(4t) - 4 \sin(4t)) + C_2 (-3 \sin(4t) + 4 \cos(4t)) \right]
\]

This is the derivative of the given function.

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Would you like further clarification on any part of the process? 

Here are some related questions:
1. How does the product rule apply when differentiating exponential and trigonometric functions?
2. What are the general rules for differentiating trigonometric functions?
3. Can you explain how to solve second-order differential equations with characteristic roots?
4. How would the solution change if the function involved different coefficients of \(t\)?
5. How does factoring out exponential terms help in simplifying expressions?

**Tip:** When dealing with trigonometric functions, it's useful to recall their standard derivatives for efficiency in solving.

Find the derivative of C1e^-3t cos(4t) + C2e^-3t sin(4t) using the product rule. This calculus problem involves differentiating exponential and trigonometric functions. Includes step-by-step solution and explanation of key formulas, such as the product rule, derivatives of exponential functions, and trigonometric derivatives. Suitable for undergraduate calculus students.

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