To derive \( \cos^2(3t) \) with respect to \( t \), we need to use the chain rule and the power rule.

Given the function \( f(t) = \cos^2(3t) \), we can rewrite it as:

\[
f(t) = \left( \cos(3t) \right)^2
\]

Now, differentiate this expression using the chain rule.

### Step 1: Differentiate the outer function
The derivative of \( u^2 \) (where \( u = \cos(3t) \)) is:

\[
\frac{d}{dt} \left( u^2 \right) = 2u \frac{du}{dt}
\]

In this case, \( u = \cos(3t) \), so we substitute back:

\[
\frac{d}{dt} \left( \cos^2(3t) \right) = 2 \cos(3t) \cdot \frac{d}{dt} \left( \cos(3t) \right)
\]

### Step 2: Differentiate the inner function
Now, we need the derivative of \( \cos(3t) \) with respect to \( t \). Using the chain rule again:

\[
\frac{d}{dt} \left( \cos(3t) \right) = -\sin(3t) \cdot \frac{d}{dt} (3t) = -3 \sin(3t)
\]

### Step 3: Combine the results
Now substitute this derivative back into the expression:

\[
\frac{d}{dt} \left( \cos^2(3t) \right) = 2 \cos(3t) \cdot \left( -3 \sin(3t) \right)
\]

Simplifying:

\[
\frac{d}{dt} \left( \cos^2(3t) \right) = -6 \cos(3t) \sin(3t)
\]

### Step 4: Use the double angle identity (optional)
We can simplify the result further using the double-angle identity for sine:

\[
\sin(2\theta) = 2\sin(\theta)\cos(\theta)
\]

Here, \( \theta = 3t \), so:

\[
-6 \cos(3t) \sin(3t) = -3 \sin(6t)
\]

Thus, the derivative of \( \cos^2(3t) \) is:

\[
\frac{d}{dt} \left( \cos^2(3t) \right) = -3 \sin(6t)
\]

Would you like further details or have any questions?

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Here are some related questions to explore further:

1. How do you apply the chain rule to trigonometric functions?
2. What is the derivative of \( \sin^2(2x) \)?
3. How do trigonometric identities simplify derivative expressions?
4. Can the product rule apply when differentiating trigonometric products?
5. What is the integral of \( \cos^2(3t) \)?

**Tip:** When working with trigonometric functions, keep the key identities (like the double angle or Pythagorean identities) in mind; they often simplify your results.

Learn how to derive cos^2(3t) step-by-step using the chain rule and trigonometric identities. This detailed solution covers the chain rule for derivatives, the power rule, and simplifies the result using the double-angle identity for sine. Suitable for college-level calculus students.

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