Let `f(t)="ln"(t)`. Then, `(d)/(dx)(int_(x^(2))^(x^(3))f(t)" dt")`

Options

A. has value 0 when x=0
B. has value 0 when x = 1 and `x=4//9`
C. has value 9 `e^(2)-4e` when x=e`
D. has differential coefficient `27e-8` for x=e

Explanation

To solve the problem, we need to differentiate the integral of the function \( f(t) = \ln(t) \) with respect to \( x \), where the limits of integration are \( x^2 \) and \( x^3 \). We will use the Leibniz rule for differentiation under the integral sign.

### Step-by-Step Solution:

1. **Identify the Function and Limits:**
We have \( f(t) = \ln(t) \) and we need to differentiate:
\[ \frac{d}{dx} \left( \int_{x^2}^{x^3} f(t) \, dt \right) \]

2. **Apply Leibniz's Rule:**
According to Leibniz's rule, if \( F(x) = \int_{g(x)}^{h(x)} f(t) \, dt \), then:
\[ \frac{dF}{dx} = f(h(x)) \cdot h'(x) - f(g(x)) \cdot g'(x) \]
Here, \( g(x) = x^2 \) and \( h(x) = x^3 \).

3. **Calculate the Derivatives of the Limits:**
- \( g'(x) = \frac{d}{dx}(x^2) = 2x \)
- \( h'(x) = \frac{d}{dx}(x^3) = 3x^2 \)

4. **Evaluate \( f(h(x)) \) and \( f(g(x)) \):**
- \( f(h(x)) = f(x^3) = \ln(x^3) = 3 \ln(x) \)
- \( f(g(x)) = f(x^2) = \ln(x^2) = 2 \ln(x) \)

5. **Substitute into Leibniz's Rule:**
Now substituting these values into the formula:
\[ \frac{d}{dx} \left( \int_{x^2}^{x^3} f(t) \, dt \right) = f(x^3) \cdot h'(x) - f(x^2) \cdot g'(x) \]
\[ = (3 \ln(x)) \cdot (3x^2) - (2 \ln(x)) \cdot (2x) \]
\[ = 9x^2 \ln(x) - 4x \ln(x) \]

6. **Combine Like Terms:**
Factor out \( \ln(x) \):
\[ = (9x^2 - 4x) \ln(x) \]

### Final Answer:
Thus, the derivative is:
\[ \frac{d}{dx} \left( \int_{x^2}^{x^3} \ln(t) \, dt \right) = (9x^2 - 4x) \ln(x) \]