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The given function is:

\[
y = \cos(\sin(3\theta))
\]

We are tasked with finding the first and second derivatives, \( y' \) and \( y'' \), with respect to \( \theta \).

### Step 1: Finding \( y' \) (First Derivative)
We will apply the **chain rule** to differentiate \( y \).

- The outer function is \( \cos(u) \) where \( u = \sin(3\theta) \).
- The derivative of \( \cos(u) \) is \( -\sin(u) \).
- Now we differentiate \( u = \sin(3\theta) \). The derivative of \( \sin(3\theta) \) is \( 3\cos(3\theta) \) (again using the chain rule).

Thus, the first derivative \( y' \) is:

\[
y' = -\sin(\sin(3\theta)) \cdot 3\cos(3\theta)
\]

### Step 2: Finding \( y'' \) (Second Derivative)
To find \( y'' \), we need to differentiate \( y' \).

\[
y' = -3\sin(\sin(3\theta))\cos(3\theta)
\]

We will again apply the product and chain rules.

- Differentiate the first term \( \sin(\sin(3\theta)) \):
  \[
  \frac{d}{d\theta}[\sin(\sin(3\theta))] = \cos(\sin(3\theta)) \cdot 3\cos(3\theta)
  \]

- Differentiate the second term \( \cos(3\theta) \):
  \[
  \frac{d}{d\theta}[\cos(3\theta)] = -3\sin(3\theta)
  \]

Now, apply the product rule to differentiate \( y' \):

\[
y'' = -3 \left[ \cos(\sin(3\theta)) \cdot 3\cos(3\theta) \cdot \cos(3\theta) - \sin(\sin(3\theta)) \cdot 3\sin(3\theta) \right]
\]

Simplifying:

\[
y'' = -9 \left[ \cos(\sin(3\theta)) \cdot \cos^2(3\theta) - \sin(\sin(3\theta)) \cdot \sin(3\theta) \right]
\]

Thus, the second derivative \( y'' \) is:

\[
y'' = -9 \left[ \cos(\sin(3\theta)) \cdot \cos^2(3\theta) - \sin(\sin(3\theta)) \cdot \sin(3\theta) \right]
\]

Let me know if you need further clarifications or details about the steps!

### 5 Related Questions:
1. How does the chain rule apply to trigonometric functions?
2. What is the general rule for differentiating composite functions?
3. How can you differentiate products of trigonometric functions using the product rule?
4. How would the differentiation process change if \( \theta \) were replaced with \( x \)?
5. What are common mistakes to avoid when applying the chain and product rules together?

**Tip:** Always remember to simplify the result as much as possible after applying the chain and product rules, as it makes the expression clearer.

y = cos(sin(3θ)). Find the first and second derivatives y' and y''.

Learn how to find the first and second derivatives of the function y = cos(sin(3θ)). This problem demonstrates the use of the chain rule and product rule for differentiating composite and trigonometric functions. Suitable for high school calculus students.

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