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The given problem is asking to calculate the derivative of an integral expression using **Part 2 of the Fundamental Theorem of Calculus**. The expression is:

\[
\frac{d}{dx} \left( \int_{x^8}^{e^{2x}} \ln(t) \, dt \right)
\]

### Part 2 of the Fundamental Theorem of Calculus (FTC):
This theorem states that if \( F(x) \) is defined as:

\[
F(x) = \int_{a(x)}^{b(x)} f(t) \, dt
\]

then the derivative of \( F(x) \) is:

\[
\frac{d}{dx} F(x) = f(b(x)) \cdot b'(x) - f(a(x)) \cdot a'(x)
\]

### Applying the theorem:

In the given problem:
- The integrand \( f(t) = \ln(t) \)
- The upper limit of integration \( b(x) = e^{2x} \)
- The lower limit of integration \( a(x) = x^8 \)

Thus, by applying the formula for the derivative of an integral with variable limits:

\[
\frac{d}{dx} \left( \int_{x^8}^{e^{2x}} \ln(t) \, dt \right)
= \ln(e^{2x}) \cdot \frac{d}{dx} (e^{2x}) - \ln(x^8) \cdot \frac{d}{dx} (x^8)
\]

Now, calculating each part:
1. \( \ln(e^{2x}) = 2x \)
2. \( \frac{d}{dx} (e^{2x}) = 2e^{2x} \)
3. \( \ln(x^8) = 8 \ln(x) \)
4. \( \frac{d}{dx} (x^8) = 8x^7 \)

Now, putting everything together:

\[
\frac{d}{dx} \left( \int_{x^8}^{e^{2x}} \ln(t) \, dt \right)
= 2x \cdot 2e^{2x} - 8 \ln(x) \cdot 8x^7
\]

Simplifying:

\[
= 4xe^{2x} - 64x^7 \ln(x)
\]

Thus, the derivative is:

\[
4xe^{2x} - 64x^7 \ln(x)
\]

Would you like further details on this process or have any questions?

Here are 5 follow-up questions:
1. How does the Fundamental Theorem of Calculus relate to differentiation of integrals?
2. Can the expression be simplified further under any conditions?
3. What happens if the limits of integration were constants instead of functions of \(x\)?
4. What is the interpretation of this derivative in terms of area under a curve?
5. How does the chain rule play a role in this calculation?

**Tip:** When applying the Fundamental Theorem of Calculus with variable limits, always differentiate the limits first before handling the integrand.

Calculate the derivative of the integral expression d/dx [∫(x^8 to e^(2x)) ln(t) dt] using Part 2 of the Fundamental Theorem of Calculus.

Learn how to calculate the derivative of an integral with variable limits using the Fundamental Theorem of Calculus Part 2. The problem involves differentiating the integral ∫(x^8 to e^(2x)) ln(t) dt with respect to x. Step-by-step solution included, suitable for college-level calculus students.

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