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The given function is:

\[
y = \cos\left(e^{-3\theta^2}\right)
\]

We are asked to find the derivative of this function with respect to \(\theta\).

### Steps for differentiation:

1. **Apply the chain rule**: Since this is a composition of functions (a cosine function with an exponential function inside), we need to apply the chain rule.

2. **Differentiate the outer function**: The derivative of \(\cos(u)\) with respect to \(u\) is \(-\sin(u)\).

   So:
   \[
   \frac{dy}{d\theta} = -\sin\left(e^{-3\theta^2}\right) \cdot \frac{d}{d\theta}\left(e^{-3\theta^2}\right)
   \]

3. **Differentiate the inner function**: Now, we differentiate \(e^{-3\theta^2}\) with respect to \(\theta\). Using the chain rule again:
   \[
   \frac{d}{d\theta}\left(e^{-3\theta^2}\right) = e^{-3\theta^2} \cdot \frac{d}{d\theta}(-3\theta^2)
   \]
   The derivative of \(-3\theta^2\) is \(-6\theta\), so:
   \[
   \frac{d}{d\theta}\left(e^{-3\theta^2}\right) = e^{-3\theta^2} \cdot (-6\theta)
   \]

4. **Combine the results**: Substituting back into the original derivative, we get:
   \[
   \frac{dy}{d\theta} = -\sin\left(e^{-3\theta^2}\right) \cdot e^{-3\theta^2} \cdot (-6\theta)
   \]

   Simplifying the expression:
   \[
   \frac{dy}{d\theta} = 6\theta \cdot \sin\left(e^{-3\theta^2}\right) \cdot e^{-3\theta^2}
   \]

Thus, the derivative is:
\[
\frac{dy}{d\theta} = 6\theta \sin\left(e^{-3\theta^2}\right) e^{-3\theta^2}
\]

Would you like further details or have any other questions?

Here are some related questions to further your understanding:
1. How do you apply the chain rule to more complex composite functions?
2. What is the derivative of \( \sin(e^{x^2}) \)?
3. How does the chain rule apply to trigonometric functions?
4. How would you handle the derivative of a product of two functions?
5. Can you apply the chain rule to integrals?

**Tip**: Always differentiate the outer function first, then work your way inward with the chain rule when dealing with composite functions.

Find the derivative of the given function: y = cos(e^(-3θ^2))

Learn how to find the derivative of the function y = cos(e^(-3θ^2)) using the chain rule. This step-by-step solution covers the differentiation of trigonometric and exponential functions, suitable for students in advanced calculus or high school Grades 11-12.

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