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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.

Suppose that f(x) = x e^x - e^x. Find f'(4).

Learn how to find the derivative of the function f(x) = x e^x - e^x and evaluate f'(4) using product rule and exponential differentiation. Step-by-step solution suitable for students in Grades 11-12.

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