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!