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Let's solve the given problem step by step.

### Problem:
You are given:
\[ y = \cos(x^6) + \cos^6(x) \]
You need to find the derivative \( y' \).

### Step 1: Derivative of \( \cos(x^6) \)
Using the chain rule:
\[
\frac{d}{dx} \left( \cos(x^6) \right) = -\sin(x^6) \cdot \frac{d}{dx}(x^6)
\]
Since \( \frac{d}{dx}(x^6) = 6x^5 \), we get:
\[
\frac{d}{dx} \left( \cos(x^6) \right) = -6x^5 \sin(x^6)
\]

### Step 2: Derivative of \( \cos^6(x) \)
Again, using the chain rule:
\[
\frac{d}{dx} \left( \cos^6(x) \right) = 6\cos^5(x) \cdot \frac{d}{dx}(\cos(x))
\]
We know that \( \frac{d}{dx}(\cos(x)) = -\sin(x) \), so:
\[
\frac{d}{dx} \left( \cos^6(x) \right) = 6\cos^5(x)(-\sin(x)) = -6\cos^5(x)\sin(x)
\]

### Step 3: Combine the results
Now, combining the derivatives of both terms:
\[
y' = -6x^5 \sin(x^6) - 6 \cos^5(x) \sin(x)
\]

### Conclusion:
The correct answer is:
\[
-6x^5 \sin(x^6) - 6 \cos^5(x) \sin(x)
\]
This corresponds to the second option.

Would you like any further details or clarifications?

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Here are 5 related questions for you to consider:
1. How does the chain rule apply in this context?
2. What would happen if the exponent in \( \cos(x^6) \) changed to 7?
3. How would you differentiate \( \sin^4(x^3) \)?
4. What is the product rule, and when do you use it?
5. How do higher-order derivatives affect the original function?

**Tip**: Always simplify the inside function first when applying the chain rule—it makes differentiating easier!

If y = cos(x^6) + cos^6(x), then y' = ....?

Learn how to differentiate the function y = cos(x^6) + cos^6(x) step by step using the chain rule. This problem involves key calculus concepts, including trigonometric differentiation and chain rule application, and is suitable for high school students in Grades 11-12.

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