Differentiate e^x cos^4x for nth derivative 

To find the \( n \)-th derivative of the function \( f(x) = e^x \cos^4(x) \), we need to understand the general pattern that emerges after applying the differentiation process repeatedly.

### Step-by-Step Process:

1. **First Derivative**:
   Let's begin by applying the product rule to differentiate \( f(x) = e^x \cos^4(x) \):
   \[
   f'(x) = e^x \cos^4(x) + e^x \frac{d}{dx}(\cos^4(x)).
   \]
   We need to calculate the derivative of \( \cos^4(x) \).

2. **Derivative of \( \cos^4(x) \)**:
   Use the chain rule:
   \[
   \frac{d}{dx}(\cos^4(x)) = 4\cos^3(x) \cdot (-\sin(x)) = -4\cos^3(x) \sin(x).
   \]

3. **First Derivative (Complete)**:
   Now substitute back:
   \[
   f'(x) = e^x \cos^4(x) - 4e^x \cos^3(x) \sin(x).
   \]
   Factor out \( e^x \):
   \[
   f'(x) = e^x (\cos^4(x) - 4\cos^3(x) \sin(x)).
   \]

4. **General Pattern**:
   Notice that \( e^x \) appears in each term, and the derivative involves differentiating powers of trigonometric functions. This suggests that after the nth derivative, the expression will be of the form:
   \[
   f^{(n)}(x) = e^x P_n(\cos(x), \sin(x)),
   \]
   where \( P_n(\cos(x), \sin(x)) \) is a polynomial in \( \cos(x) \) and \( \sin(x) \) with terms that depend on \( n \). The structure of the polynomial will involve alternating between products of \( \cos(x) \) and \( \sin(x) \), and the powers will evolve as a result of repeated differentiation.

### Conclusion:
The nth derivative will take the form:
\[
f^{(n)}(x) = e^x P_n(\cos(x), \sin(x)),
\]
where \( P_n(\cos(x), \sin(x)) \) is a polynomial generated through repeated product rule and chain rule applications.

Would you like more details on the general form of \( P_n(\cos(x), \sin(x)) \) or the derivation for specific small \( n \) values?

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### 5 Related Questions:
1. How does the Leibniz rule for the nth derivative of a product apply in this case?
2. How can we compute the nth derivative of \( \cos^n(x) \) in isolation?
3. What role does the exponential factor \( e^x \) play in simplifying the derivative process?
4. How does trigonometric differentiation affect the pattern in successive derivatives?
5. Can we use series expansion to approximate the nth derivative of \( e^x \cos^4(x) \)?

**Tip**: When differentiating products of functions, always look for patterns in the terms produced by applying the product rule recursively.

Differentiate e^x cos^4x for nth derivative

Explore the step-by-step process for finding the nth derivative of the function e^x cos^4(x). This guide covers differentiation techniques including the product rule, chain rule, and the Leibniz rule for the nth derivative, with applications to trigonometric and exponential functions. Suitable for advanced high school students and beyond.

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