Let's solve this step by step.

Problem Analysis:

Given: $f(x) = \int_{4}^{x} t^8 \, dt$

1. Find $f'(x)$: According to the Fundamental Theorem of Calculus, if: $F(x) = \int_{a}^{x} g(t) \, dt$ then: $F'(x) = g(x)$ So here, $f'(x) = x^8$.

2. Evaluate $f'(5)$: Now, substitute $x = 5$: $f'(5) = 5^8$

Let's compute $5^8$:

$5^8 = 390625$

Answers:

1. $f'(x) = x^8$
2. $f'(5) = 390625$

Explanation of Mistakes:

• You initially wrote $t^8$ instead of $x^8$ for $f'(x)$. The derivative should be evaluated with respect to $x$.
• For $f'(5)$, you provided $3125$, which is incorrect. You computed $5^5$ instead of $5^8$.

Would you like a more detailed explanation or have any specific questions? Here are related questions you might find useful:

1. What is the Fundamental Theorem of Calculus?
2. How do you differentiate an integral function?
3. What is the significance of changing the variable from $t$ to $x$?
4. Can the bounds of integration affect the derivative?
5. How can we simplify calculations like $5^8$ without a calculator?

Tip: When differentiating an integral with variable bounds, remember to apply the Fundamental Theorem of Calculus and use the upper bound directly.