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The problem asks for \( f'(x) \) and \( f'(3) \) based on the integral definition of \( f(x) \), given by:

\[
f(x) = \int_5^x t^8 \, dt
\]

### Step 1: Use the Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus, if we have:

\[
f(x) = \int_a^x g(t) \, dt
\]

then the derivative \( f'(x) \) is simply:

\[
f'(x) = g(x)
\]

In this case, \( g(t) = t^8 \), so:

\[
f'(x) = x^8
\]

### Step 2: Calculate \( f'(3) \)

Now, we substitute \( x = 3 \) into the expression for \( f'(x) \):

\[
f'(3) = 3^8 = 6561
\]

### Final Answers:

- \( f'(x) = x^8 \)
- \( f'(3) = 6561 \)

Would you like further details or have any questions?

### Related Questions:
1. How does the Fundamental Theorem of Calculus apply to definite integrals?
2. What are the conditions under which you can use the Fundamental Theorem of Calculus?
3. How would \( f'(x) \) change if the exponent on \( t \) were different (e.g., \( t^5 \))?
4. How do we compute the second derivative, \( f''(x) \), in this case?
5. What happens if we change the lower limit of the integral from 5 to some other value?

### Tip:
Always remember that the derivative of an integral with a variable upper limit is the integrand evaluated at that upper limit, according to the Fundamental Theorem of Calculus.

If f(x) = ∫ from 5 to x of t^8 dt, find f'(x) and f'(3).

Learn how to apply the Fundamental Theorem of Calculus to find the derivative of the function f(x) = ∫ from 5 to x of t^8 dt. The problem also calculates f'(3), demonstrating key differentiation techniques. Suitable for advanced high school or college-level calculus students.

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