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Solve using the product rule with step by step steps

To solve the given problem using the product rule, we'll go step by step. The function given is:

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

### Step 1: Expand the function first
We will simplify the expression before applying the product rule. Notice that \((\cos x + \sin x)(\cos x - \sin x)\) is a difference of squares:

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

So the function simplifies to:

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

### Step 2: Differentiate the simplified function
Now, we will differentiate \( y = 6(\cos^2 x - \sin^2 x) \) with respect to \( x \). We can apply the chain rule for each term:

- The derivative of \( \cos^2 x \) is \( 2\cos x(-\sin x) = -2\cos x \sin x \)
- The derivative of \( \sin^2 x \) is \( 2\sin x \cos x = 2\sin x \cos x \)

Thus:

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

### Step 3: Simplify the derivative
\[
\frac{dy}{dx} = 6\left( -4\cos x \sin x \right)
\]

So the final result is:

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

### Summary:
The derivative of the given function \( y = 6(\cos x + \sin x)(\cos x - \sin x) \) is:

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

Would you like any further clarifications or a deeper explanation?

Here are five questions to expand on this:
1. What is the significance of the difference of squares in trigonometric identities?
2. How does the product rule apply when the function is already simplified?
3. Can you verify the result using a different method, such as implicit differentiation?
4. What is the geometric interpretation of this derivative for specific values of \( x \)?
5. How can you further simplify or interpret \( -24\cos x \sin x \) using known trigonometric identities?

**Tip**: Always look out for algebraic simplifications like difference of squares before applying differentiation techniques like the product rule.

Find dy/dx for y = 6(cos x + sin x)(cos x - sin x) using the product rule with step-by-step steps.

This page explains how to find the derivative of the function y = 6(cos x + sin x)(cos x - sin x) using the product rule and trigonometric identities. It includes a step-by-step breakdown of the process, starting with the difference of squares identity and applying the chain rule. Suitable for Calculus students.

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