To differentiate \( y = 3x + x \tan(x) \) with respect to \( x \), we will apply the rules of differentiation.

### Step-by-Step Solution:
1. The derivative of \( 3x \) with respect to \( x \) is:
   \[
   \frac{d}{dx}(3x) = 3
   \]

2. Now, we differentiate \( x \tan(x) \). This requires the **product rule**:
   - The product rule states that if \( u(x) = x \) and \( v(x) = \tan(x) \), then:
     \[
     \frac{d}{dx}(x \tan(x)) = \frac{d}{dx}(x) \cdot \tan(x) + x \cdot \frac{d}{dx}(\tan(x))
     \]
     - The derivative of \( x \) is 1.
     - The derivative of \( \tan(x) \) is \( \sec^2(x) \).

   Therefore:
   \[
   \frac{d}{dx}(x \tan(x)) = 1 \cdot \tan(x) + x \cdot \sec^2(x) = \tan(x) + x \sec^2(x)
   \]

### Final result:
Putting it all together, the derivative of \( y \) with respect to \( x \) is:
\[
\frac{dy}{dx} = 3 + \tan(x) + x \sec^2(x)
\]

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Would you like more details or have any questions about this?

Here are five related questions to deepen your understanding:
1. How does the product rule apply in other trigonometric functions?
2. What is the derivative of \( x^2 \tan(x) \)?
3. How do you differentiate a product involving \( \cot(x) \) instead of \( \tan(x) \)?
4. What are some common uses of the derivative of \( \tan(x) \) in physics?
5. Can you find the second derivative \( \frac{d^2y}{dx^2} \) for this equation?

**Tip:** Always remember the product rule and chain rule when differentiating complex functions with products or compositions!

Learn how to differentiate the function y = 3x + x tan(x) using the product rule and trigonometric differentiation. This step-by-step solution explains key concepts such as the product rule and the derivative of tan(x). Suitable for students in Grades 11-12 studying calculus.

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