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.