To determine which of the given functions is differentiable in the interval \((-1, 2)\), we will analyze each function using the properties of differentiability and the Fundamental Theorem of Calculus.
### Step-by-Step Solution:
1. **Identify the Functions**:
Let's denote the functions provided in the problem as \( f_1(x) \), \( f_2(x) \), and \( f_3(x) \). We need to analyze each of these functions for differentiability in the interval \((-1, 2)\).
2. **Check the First Function**:
- **Function**: \( f_1(x) = \int_{x}^{2x} \log(t^2) \, dt \)
- **Differentiability**: The logarithm function \( \log(t^2) \) is not defined for \( t \leq 0 \). Since \( t \) ranges from \( x \) to \( 2x \), if \( x \) is negative (which it is in the interval \((-1, 0)\)), the function is not differentiable in that interval.
- **Conclusion**: \( f_1(x) \) is **not differentiable** in \((-1, 2)\).
3. **Check the Second Function**:
- **Function**: \( f_2(x) = \int_{x}^{2x} \sin(t) \, dt \)
- **Differentiability**: The sine function is defined for all real numbers, so we can differentiate this function using the Fundamental Theorem of Calculus.
- **Differentiation**:
\[
f_2'(x) = \sin(2x) \cdot 2 - \sin(x) \cdot 1
\]
- Since both \( \sin(2x) \) and \( \sin(x) \) are defined for all \( x \), \( f_2(x) \) is differentiable in \((-1, 2)\).
- **Conclusion**: \( f_2(x) \) is **differentiable** in \((-1, 2)\).
4. **Check the Third Function**:
- **Function**: \( f_3(x) = \int_{1}^{2x} \frac{1 - t + t^2}{1 + t + t^2} \, dt \)
- **Differentiability**: The integrand \( \frac{1 - t + t^2}{1 + t + t^2} \) is a rational function and is defined for all \( t \) (the denominator does not equal zero for real \( t \)).
- **Differentiation**:
\[
f_3'(x) = \frac{1 - (2x) + (2x)^2}{1 + (2x) + (2x)^2} \cdot 2
\]
- This derivative is also defined for all \( x \) in \((-1, 2)\).
- **Conclusion**: \( f_3(x) \) is **differentiable** in \((-1, 2)\).
### Final Conclusion:
Among the functions analyzed:
- \( f_1(x) \) is **not differentiable** in \((-1, 2)\).
- \( f_2(x) \) is **differentiable** in \((-1, 2)\).
- \( f_3(x) \) is **differentiable** in \((-1, 2)\).
Thus, the answer to the question "Which of the following is differentiable in the interval \((-1, 2)\)?" is **\( f_2(x) \) and \( f_3(x) \)**.
