`f(x) = sin^(-1)(sin x)` then `(d)/(dx)f(x)` at `x = (7pi)/(2)` is

Options

A. 1
B. `-1`
C. Not differentiable
D. None of these

Explanation

To solve the problem, we need to find the derivative of the function \( f(x) = \sin^{-1}(\sin x) \) at the point \( x = \frac{7\pi}{2} \).

### Step-by-Step Solution:

1. **Understanding the Function**:
The function \( f(x) = \sin^{-1}(\sin x) \) is defined as the inverse sine of the sine of \( x \). However, the output of \( \sin^{-1}(y) \) is restricted to the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \).

2. **Identifying the Range of \( x \)**:
The sine function is periodic with a period of \( 2\pi \). Therefore, we can express \( x \) in terms of its equivalent angle within the principal range of the sine function.

3. **Finding the Equivalent Angle**:
For \( x = \frac{7\pi}{2} \):
\[ \frac{7\pi}{2} - 3\pi = \frac{7\pi}{2} - \frac{6\pi}{2} = \frac{\pi}{2} \]
Thus, \( \sin\left(\frac{7\pi}{2}\right) = \sin\left(\frac{\pi}{2}\right) = 1 \).

4. **Evaluating \( f(x) \)**:
Therefore, we have:
\[ f\left(\frac{7\pi}{2}\right) = \sin^{-1}(1) = \frac{\pi}{2} \]

5. **Finding the Derivative**:
To find the derivative \( f'(x) \), we need to consider the piecewise nature of \( f(x) \):
- For \( x \) in the interval \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), \( f(x) = x \).
- For \( x \) in the interval \( \left[\frac{\pi}{2}, \frac{3\pi}{2}\right] \), \( f(x) = \pi - x \).
- For \( x \) in the interval \( \left[\frac{3\pi}{2}, \frac{5\pi}{2}\right] \), \( f(x) = x - 2\pi \).
- For \( x \) in the interval \( \left[\frac{5\pi}{2}, \frac{7\pi}{2}\right] \), \( f(x) = 3\pi - x \).
- For \( x \) in the interval \( \left[\frac{7\pi}{2}, \frac{9\pi}{2}\right] \), \( f(x) = x - 4\pi \).

6. **Calculating Left-Hand and Right-Hand Derivatives**:
- **Left-Hand Derivative** at \( x = \frac{7\pi}{2} \):
\[ f(x) = 3\pi - x \quad \text{for } x < \frac{7\pi}{2} \]
\[ f'(x) = -1 \quad \text{(derivative of } 3\pi - x\text{)} \]

- **Right-Hand Derivative** at \( x = \frac{7\pi}{2} \):
\[ f(x) = x - 4\pi \quad \text{for } x > \frac{7\pi}{2} \]
\[ f'(x) = 1 \quad \text{(derivative of } x - 4\pi\text{)} \]

7. **Conclusion**:
Since the left-hand derivative (\(-1\)) is not equal to the right-hand derivative (\(1\)), the function \( f(x) \) is not differentiable at \( x = \frac{7\pi}{2} \).

### Final Answer:
The derivative \( \frac{d}{dx} f(x) \) at \( x = \frac{7\pi}{2} \) is **not differentiable**.