https://file.aiquickdraw.com/mathbot/file/1728490121977ahmlgrfj.jpg

The function given in the image is \( f(x) = x^7 e^x \), and you're asked to find \( f'(1) \), which means we need to compute the derivative of the function \( f(x) \) and then evaluate it at \( x = 1 \).

### Step 1: Differentiate the function
We use the **product rule** because \( f(x) = x^7 e^x \) is the product of two functions:
\[
f(x) = u(x) \cdot v(x)
\]
where:
- \( u(x) = x^7 \)
- \( v(x) = e^x \)

The product rule states:
\[
(f \cdot g)' = f' g + f g'
\]

#### First, differentiate \( u(x) = x^7 \):
\[
u'(x) = 7x^6
\]

#### Then, differentiate \( v(x) = e^x \):
\[
v'(x) = e^x
\]

Now, apply the product rule:
\[
f'(x) = u'(x)v(x) + u(x)v'(x)
\]
\[
f'(x) = 7x^6 e^x + x^7 e^x
\]

### Step 2: Evaluate \( f'(x) \) at \( x = 1 \)
Now, substitute \( x = 1 \) into the derivative:
\[
f'(1) = 7(1)^6 e^1 + (1)^7 e^1
\]
\[
f'(1) = 7e + e
\]
\[
f'(1) = 8e
\]

So, the answer is:
\[
f'(1) = 8e
\]

Would you like more details or have any questions about this process? 

Here are 5 related questions to expand your understanding:
1. How is the product rule used to differentiate functions?
2. What is the derivative of \( e^x \), and why is it the same?
3. Can the chain rule be combined with the product rule in similar problems?
4. What is the second derivative \( f''(x) \) for this function?
5. How would you compute the derivative if the power of \( x \) was a function of \( x \)?

**Tip:** Always check whether a function is a product of two functions when differentiating—it can simplify the problem using the product rule.

This problem demonstrates the use of the product rule to find the derivative of f(x) = x^7 e^x and evaluates it at x = 1. The solution walks through the step-by-step differentiation process, suitable for high school or college calculus students.

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