To find the derivative of \( y = \cos^2(x) \), we can approach this using the **chain rule**.

### Step 1: Rewrite the function
The given function can be rewritten as:

\[
y = (\cos(x))^2
\]

This shows the function is a composite function, i.e., an outer function \( u(x) = x^2 \) and an inner function \( v(x) = \cos(x) \).

### Step 2: Apply the chain rule
The **chain rule** states that if \( y = f(g(x)) \), then:

\[
\frac{dy}{dx} = f'(g(x)) \cdot g'(x)
\]

Here:

- The outer function is \( f(u) = u^2 \), and its derivative is \( f'(u) = 2u \).
- The inner function is \( v(x) = \cos(x) \), and its derivative is \( v'(x) = -\sin(x) \).

Now, applying the chain rule:

\[
\frac{dy}{dx} = 2 \cos(x) \cdot (-\sin(x))
\]

### Step 3: Simplify
This simplifies to:

\[
\frac{dy}{dx} = -2 \cos(x) \sin(x)
\]

This can also be written as:

\[
\frac{dy}{dx} = -\sin(2x)
\]

(using the double-angle identity \( \sin(2x) = 2 \sin(x) \cos(x) \)).

### Conclusion:
The derivative of \( y = \cos^2(x) \) is:

\[
\frac{dy}{dx} = -\sin(2x)
\]

Let me know if you'd like any further clarifications!

Here are 5 related questions:
1. What is the second derivative of \( y = \cos^2(x) \)?
2. How does the derivative change if \( y = \sin^2(x) \)?
3. Can you find the derivative of \( y = \cos^3(x) \)?
4. How do you find the derivative of \( y = \cos(2x) \)?
5. How does the chain rule apply to more complex trigonometric functions?

**Tip:** The chain rule is useful when differentiating composite functions, especially when functions involve powers of trigonometric functions.

Learn how to find the derivative of y = cos²(x) using the chain rule. This step-by-step guide covers key concepts like trigonometric differentiation and the double-angle identity, ideal for students in Grades 10-12.

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