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The problem asks to calculate the derivative:

\[
\frac{d}{dx} \left( \int_{x^4}^{e^{11x}} \ln(t) \, dt \right)
\]

using Part 2 of the Fundamental Theorem of Calculus (FTC).

### Part 2 of the Fundamental Theorem of Calculus:
It states that if \( F(x) = \int_{a(x)}^{b(x)} f(t) \, dt \), then

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

where \( f(t) \) is the integrand, \( a(x) \) and \( b(x) \) are the lower and upper limits of integration respectively, and \( f'(x) \) is the derivative of the limits with respect to \( x \).

### Step-by-Step Solution:

Here, \( f(t) = \ln(t) \), \( a(x) = x^4 \), and \( b(x) = e^{11x} \).

So, applying the Fundamental Theorem:

\[
\frac{d}{dx} \left( \int_{x^4}^{e^{11x}} \ln(t) \, dt \right) = \ln(e^{11x}) \cdot \frac{d}{dx}(e^{11x}) - \ln(x^4) \cdot \frac{d}{dx}(x^4)
\]

1. **For the upper limit** \( b(x) = e^{11x} \):
   \[
   \ln(e^{11x}) = 11x \quad \text{(since \( \ln(e^u) = u \))}
   \]
   and
   \[
   \frac{d}{dx}(e^{11x}) = 11e^{11x}.
   \]

2. **For the lower limit** \( a(x) = x^4 \):
   \[
   \ln(x^4) = 4 \ln(x)
   \]
   and
   \[
   \frac{d}{dx}(x^4) = 4x^3.
   \]

### Final Result:
Now substitute these into the formula:

\[
\frac{d}{dx} \left( \int_{x^4}^{e^{11x}} \ln(t) \, dt \right) = 11x \cdot 11e^{11x} - 4 \ln(x) \cdot 4x^3
\]

Simplifying:

\[
= 121x e^{11x} - 16x^3 \ln(x)
\]

Thus, the derivative is:

\[
\boxed{121x e^{11x} - 16x^3 \ln(x)}
\]

Would you like any further details on the solution?

### Related Questions:
1. How does the Fundamental Theorem of Calculus help in differentiating integrals?
2. What happens if one of the limits of integration is a constant rather than a function of \( x \)?
3. How would the result change if the integrand were a different function, such as \( \frac{1}{t} \)?
4. Can this method be applied if the integrand involves multiple variables, such as \( \ln(t + x) \)?
5. How would we approach the problem if the limits were swapped, i.e., \( \int_{e^{11x}}^{x^4} \ln(t) dt \)?

### Tip:
Whenever you differentiate an integral with variable limits, apply the Fundamental Theorem of Calculus carefully, remembering to differentiate both the upper and lower limits separately.

Calculate the derivative \( \frac{d}{dx} \int_{x^4}^{e^{11x}} \ln(t) dt \) using Part 2 of the Fundamental Theorem of Calculus.

Learn how to differentiate an integral with variable limits using Part 2 of the Fundamental Theorem of Calculus. This calculus problem involves finding the derivative of the integral \( \int_{x^4}^{e^{11x}} \ln(t) dt \) with respect to \( x \), applying techniques of differentiation and integration.

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