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.

Learn how to differentiate the function 7xe^x step-by-step using the product rule. This solution includes applying key calculus concepts such as the product rule and the derivative of exponential functions. Suitable for high school or college students studying calculus.

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