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To solve the derivative \(\frac{dy}{dx}\) for the given function without using the chain rule, we can simplify the expression first.

The function given is:

\[
y = 6 (\cos x + \sin x)(\cos x - \sin x)
\]

### Step 1: Simplify the Expression

Notice that \((\cos x + \sin x)(\cos x - \sin x)\) is a difference of squares:

\[
(\cos x + \sin x)(\cos x - \sin x) = \cos^2 x - \sin^2 x
\]

Thus, the equation becomes:

\[
y = 6 (\cos^2 x - \sin^2 x)
\]

### Step 2: Differentiate the Simplified Expression

Now, differentiate \(y = 6 (\cos^2 x - \sin^2 x)\) with respect to \(x\).

\[
\frac{dy}{dx} = 6 \cdot \frac{d}{dx}(\cos^2 x - \sin^2 x)
\]

### Step 3: Differentiate Each Term

The derivative of \(\cos^2 x\) is:

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

The derivative of \(\sin^2 x\) is:

\[
\frac{d}{dx}(\sin^2 x) = 2 \sin x \cdot \cos x = 2 \sin x \cos x
\]

### Step 4: Combine the Results

Now substitute these derivatives into the equation:

\[
\frac{dy}{dx} = 6 \left( -2 \cos x \sin x - 2 \sin x \cos x \right)
\]

Simplify:

\[
\frac{dy}{dx} = 6 \left( -4 \cos x \sin x \right)
\]

Thus, the final answer is:

\[
\frac{dy}{dx} = -24 \cos x \sin x
\]

Would you like more details or have any questions?

### Related Questions:
1. How can we use the chain rule to differentiate this expression?
2. What is the geometric interpretation of \(\cos^2 x - \sin^2 x\)?
3. How would the answer change if the expression was \(6(\sin x + \cos x)^2\)?
4. What are some real-world applications of this type of trigonometric identity?
5. How does the product rule relate to the approach we just used?

### Tip:
When simplifying trigonometric expressions, always look for common identities such as the difference of squares, which can make differentiation easier.

Find dy/dx for y = 6(cos(x) + sin(x))(cos(x) - sin(x)) without using the chain rule.

Find the derivative of y = 6(cos(x) + sin(x))(cos(x) - sin(x)) without using the chain rule. This problem involves trigonometric identities, simplification, and differentiation rules, suitable for high school calculus students.

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