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You provided the function:

\[
y = x \cos(x) \sin(x)
\]

To differentiate this function with respect to \( x \), we'll use the product rule and the chain rule.

### Steps:

1. **Recall the product rule**: 
   If \( u(x) \cdot v(x) \) is a product of two functions, its derivative is given by:
   \[
   \frac{d}{dx}(u(x) \cdot v(x)) = u'(x)v(x) + u(x)v'(x)
   \]

2. **Define the components**:
   Here, \( u(x) = x \) and \( v(x) = \cos(x) \sin(x) \). We need to differentiate both parts.

3. **Differentiate \( u(x) = x \)**:
   \[
   u'(x) = 1
   \]

4. **Differentiate \( v(x) = \cos(x) \sin(x) \)**:
   To differentiate this, apply the product rule again:
   \[
   \frac{d}{dx}(\cos(x) \sin(x)) = \cos(x) \cdot \frac{d}{dx}(\sin(x)) + \sin(x) \cdot \frac{d}{dx}(\cos(x))
   \]
   Using the derivatives:
   \[
   \frac{d}{dx}(\sin(x)) = \cos(x) \quad \text{and} \quad \frac{d}{dx}(\cos(x)) = -\sin(x)
   \]
   We get:
   \[
   \frac{d}{dx}(\cos(x) \sin(x)) = \cos(x) \cdot \cos(x) + \sin(x) \cdot (-\sin(x)) = \cos^2(x) - \sin^2(x)
   \]

5. **Apply the product rule**:
   Now differentiate the whole function:
   \[
   \frac{d}{dx}(x \cos(x) \sin(x)) = \frac{d}{dx}(x) \cdot \cos(x) \sin(x) + x \cdot \frac{d}{dx}(\cos(x) \sin(x))
   \]
   Substituting the derivatives:
   \[
   = 1 \cdot \cos(x) \sin(x) + x \cdot (\cos^2(x) - \sin^2(x))
   \]

Thus, the derivative is:

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

If you have any questions or want further details, feel free to ask!

### Related Questions:
1. How does the product rule work in more complex functions?
2. What is the geometrical interpretation of this derivative?
3. How does the chain rule apply to trigonometric functions?
4. Can this derivative be simplified further using trigonometric identities?
5. What is the second derivative of this function?

**Tip**: The identity \(\cos(2x) = \cos^2(x) - \sin^2(x)\) could simplify expressions with \(\cos^2(x) - \sin^2(x)\).

Learn how to differentiate the function y = x * cos(x) * sin(x) using the product and chain rules. This detailed step-by-step solution explains key calculus and trigonometric concepts, including applying the product rule, differentiating trigonometric functions, and simplifying the result.

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