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.