To differentiate the function $7x e^x$ with respect to $x$, we'll use the product rule.

The product rule states that if you have two functions $f(x)$ and $g(x)$, their derivative is:

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

Here, $f(x) = 7x$ and $g(x) = e^x$.

1. Derivative of $f(x) = 7x$ is $f'(x) = 7$.
2. Derivative of $g(x) = e^x$ is $g'(x) = e^x$ (since $e^x$ is its own derivative).

Now, apply the product rule:

$\frac{d}{dx} [7x e^x] = 7 \cdot e^x + 7x \cdot e^x$

This simplifies to:

$\frac{d}{dx} [7x e^x] = 7e^x(1 + x)$

So, the derivative is:

$\boxed{7e^x(1 + x)}$

Would you like more details on any step?

Here are 5 related questions to explore:

1. How do you apply the product rule for differentiating more complex products?
2. What is the derivative of a general function $x e^x$?
3. Can you find the second derivative of $7x e^x$?
4. How does the chain rule work when differentiating functions involving $e^x$?
5. What are some applications of derivatives involving exponential functions?

Tip: Exponential functions like $e^x$ have the unique property that they are equal to their own derivative. This simplifies many problems involving them.