To find the nth derivative of the function $y = e^x \cos^4(x)$, we'll need to differentiate the product of these two functions multiple times.

Step 1: First derivative

We use the product rule for differentiation: $\frac{d}{dx}(uv) = u'v + uv'$ where $u = e^x$ and $v = \cos^4(x)$.

First, differentiate each term:

• The derivative of $e^x$ is $e^x$.
• To differentiate $\cos^4(x)$, apply the chain rule: $\frac{d}{dx}(\cos^4(x)) = 4\cos^3(x) \cdot (-\sin(x)) = -4\cos^3(x) \sin(x).$

Now, apply the product rule: $y' = e^x \cdot (-4\cos^3(x) \sin(x)) + e^x \cos^4(x).$

Factor out $e^x$: $y' = e^x [-4\cos^3(x) \sin(x) + \cos^4(x)].$

Step 2: Second derivative

For the second derivative, differentiate the first derivative again using the product rule. Let $u = e^x$ and $v = -4\cos^3(x) \sin(x) + \cos^4(x)$. You'll need to differentiate the trigonometric terms accordingly.

General Approach for nth derivative

Each differentiation will increase the complexity of the expression, involving terms with higher powers of $\sin(x)$ and $\cos(x)$ and possibly products of these with the exponential term $e^x$. The nth derivative will follow a pattern involving the product rule applied repeatedly, and will alternate in signs and powers of trigonometric functions, with exponential growth due to $e^x$.

Conclusion:

The nth derivative of $y = e^x \cos^4(x)$ involves repeated applications of the product rule. The resulting pattern will always have $e^x$ factored out, with increasingly complex trigonometric terms involving powers of $\cos(x)$ and $\sin(x)$, and coefficients that come from the differentiation process.

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Would you like a deeper dive into the second derivative or more on the general nth term? Feel free to ask!

Here are 5 related questions to expand on this topic:

1. What is the second derivative of $y = e^x \cos^4(x)$?
2. How does the Leibniz rule help in differentiating products multiple times?
3. What is the pattern of coefficients in the nth derivative of $e^x \cos^n(x)$?
4. Can we generalize the nth derivative for any product of exponential and trigonometric functions?
5. How can Taylor series be used to approximate $e^x \cos^4(x)$?

Tip: When differentiating products, use the product rule and chain rule step by step to avoid errors, especially when dealing with higher derivatives.