To find the derivative of the function

\[
y = 1 + 2x + x e^x
\]

we differentiate term by term using basic differentiation rules, including the product rule for the term \( x e^x \).

### Step 1: Differentiate each term
- The derivative of the constant \( 1 \) is \( 0 \).
- The derivative of \( 2x \) is \( 2 \).
- The term \( x e^x \) requires the product rule because it is the product of two functions of \( x \): \( x \) and \( e^x \).

The product rule states that if \( y = u(x)v(x) \), then:

\[
\frac{d}{dx} [u(x)v(x)] = u'(x)v(x) + u(x)v'(x)
\]

Here, \( u(x) = x \) and \( v(x) = e^x \), so:

\[
\frac{d}{dx} [x e^x] = \frac{d}{dx} [x] \cdot e^x + x \cdot \frac{d}{dx} [e^x]
\]

This becomes:

\[
\frac{d}{dx} [x e^x] = 1 \cdot e^x + x \cdot e^x = e^x + x e^x
\]

### Step 2: Combine the results
Now, differentiate the entire function:

\[
\frac{dy}{dx} = \frac{d}{dx} \left( 1 + 2x + x e^x \right)
\]
\[
\frac{dy}{dx} = 0 + 2 + (e^x + x e^x)
\]
\[
\frac{dy}{dx} = 2 + e^x + x e^x
\]

### Conclusion:
The derivative of the function \( y = 1 + 2x + x e^x \) is:

\[
\frac{dy}{dx} = 2 + e^x + x e^x
\]

Would you like more details or further clarifications?

Here are 5 related questions:
1. How would the derivative change if the function had \( e^{2x} \) instead of \( e^x \)?
2. What is the second derivative of \( y = 1 + 2x + x e^x \)?
3. How can you apply the product rule to more complex products?
4. How do you find the derivative of the function \( y = x^2 e^x \)?
5. What is the derivative of \( e^x \) divided by \( x \)?

**Tip:** When differentiating products of functions, always apply the product rule to avoid missing terms.

Learn how to find the derivative of the function y = 1 + 2x + x e^x using basic differentiation rules and the product rule. This problem involves calculus concepts, including the differentiation of exponential functions. Suitable for Grades 11-12 and college-level students.

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