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.