The problem asks to find $f'(4)$, where the function $f(x)$ is defined as:

$f(x) = x e^x - e^x$

Step 1: Differentiate $f(x)$

We can differentiate $f(x) = x e^x - e^x$ using the product rule and the derivative of exponential functions.

1. The derivative of the first term $x e^x$ is calculated using the product rule: [ \frac{d}{dx}[x e^x] = \frac{d}{dx}[x] \cdot e^x + x \cdot \frac{d}{dx}[e^x] = e^x + x e^x ]

2. The derivative of the second term $-e^x$ is simply: $\frac{d}{dx}[-e^x] = -e^x$

So, the derivative of $f(x)$ is: $f'(x) = (e^x + x e^x) - e^x$

Simplifying: $f'(x) = x e^x$

Step 2: Evaluate $f'(4)$

Now we substitute $x = 4$ into the derivative $f'(x) = x e^x$: $f'(4) = 4 e^4$

Step 3: Approximate the value

We can compute the numerical value of $e^4 \approx 54.598$, so: $f'(4) = 4 \times 54.598 \approx 218.392$

Thus, $f'(4) \approx 218.392$.

Would you like further details on any steps? Here are some related questions you might explore:

1. How do you apply the product rule in differentiation?
2. Why does the derivative of $e^x$ remain $e^x$?
3. What are common mistakes to avoid when differentiating functions involving products?
4. How can you calculate $e^x$ without a calculator for specific values of $x$?
5. What are other examples of functions that involve products of exponential terms?

Tip: Always simplify your derivative expression before plugging in specific values. This reduces the chance of mistakes and makes calculations easier.