To find the derivative of \((x + 1)e^x\), we will use the **product rule** since the expression is a product of two functions: \( (x + 1) \) and \( e^x \).

The product rule states that:

\[
\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
\]

Let:
- \( f(x) = (x + 1) \)
- \( g(x) = e^x \)

Now, calculate the derivatives:
- \( f'(x) = \frac{d}{dx}(x + 1) = 1 \)
- \( g'(x) = \frac{d}{dx}(e^x) = e^x \)

Now apply the product rule:
\[
\frac{d}{dx}[(x + 1)e^x] = f'(x)g(x) + f(x)g'(x)
\]
\[
= 1 \cdot e^x + (x + 1) \cdot e^x
\]
\[
= e^x + (x + 1)e^x
\]

Factor out \( e^x \):
\[
= e^x(1 + x + 1)
\]
\[
= e^x(x + 2)
\]

So, the derivative of \((x + 1)e^x\) is:
\[
e^x(x + 2)
\]

Would you like more details or have any questions?

Here are 5 related questions to further explore this topic:
1. How would the derivative change if the expression was \((x + 2)e^x\)?
2. What is the second derivative of \((x + 1)e^x\)?
3. How do you apply the product rule to three or more functions?
4. Can you use the quotient rule to differentiate \(\frac{(x+1)}{e^x}\)?
5. How do you find the derivative of \(e^{x^2}\)?

**Tip:** Always check if factoring the final expression simplifies your derivative for a cleaner result!

Discover how to find the derivative of (x+1)e^x using the product rule. This solution explains step-by-step differentiation and applies fundamental calculus concepts such as the product rule. Ideal for students in Grades 11-12 or early college.

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