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To find the derivative of the given function \( f(x) = x^4 \cos(3x) \), we will use the **product rule** of differentiation. 

The product rule states that if \( f(x) = u(x) \cdot v(x) \), then:
\[
f'(x) = u'(x) v(x) + u(x) v'(x)
\]
Here, \( u(x) = x^4 \) and \( v(x) = \cos(3x) \).

### Steps:
1. Differentiate \( u(x) = x^4 \):
   \[
   u'(x) = 4x^3
   \]

2. Differentiate \( v(x) = \cos(3x) \) using the chain rule:
   \[
   v'(x) = -\sin(3x) \cdot 3 = -3\sin(3x)
   \]

Now, apply the product rule:
\[
f'(x) = u'(x) v(x) + u(x) v'(x)
\]
Substitute the values:
\[
f'(x) = 4x^3 \cos(3x) + x^4 (-3\sin(3x))
\]
Simplifying the expression:
\[
f'(x) = 4x^3 \cos(3x) - 3x^4 \sin(3x)
\]

Thus, the derivative is:
\[
f'(x) = 4x^3 \cos(3x) - 3x^4 \sin(3x)
\]

Would you like any further details or clarifications on this solution?

Here are 5 related questions to further explore:
1. How would the derivative change if the exponent of \(x\) were different, e.g., \(x^3\)?
2. What is the second derivative of this function?
3. How can we apply the chain rule in more complex trigonometric functions?
4. Can you identify a real-world situation where this type of derivative might be useful?
5. How would the derivative look if we had \( \sin(3x) \) instead of \( \cos(3x) \)?

**Tip:** When using the product rule, always make sure to differentiate both functions correctly, especially if one of them involves the chain rule.

Find the derivative of the function given below: f(x) = x^4 cos(3x)

Explore how to find the derivative of x^4 cos(3x) using calculus techniques such as the product and chain rules. This guide provides step-by-step solutions to mastering differentiation with complex functions, suitable for college-level students.

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