If `y=cos^(-1)(cosx),t h e n(dy)/(dx)` is equal to `x/y` (b) `y/(x^2)` `(x^2-y^2)/(x^2+y^2)` (d) `y/x`

Options

A. 1
B. -1
C. `(1)/(sqrt(2))`
D. none of these

Explanation

To solve the problem, we need to differentiate the function \( y = \cos^{-1}(\cos x) \) and find \( \frac{dy}{dx} \) at \( x = \frac{\pi}{4} \).

### Step-by-Step Solution:

1. **Understanding the Function**:
The function \( y = \cos^{-1}(\cos x) \) can be simplified based on the range of \( x \).
- For \( x \) in the interval \( [0, \pi] \), \( y = x \).
- For \( x \) in the interval \( [\pi, 2\pi] \), \( y = 2\pi - x \).

2. **Determine the Relevant Interval**:
Since we are interested in \( x = \frac{\pi}{4} \), which lies in the interval \( [0, \pi] \), we can use the first case:
\[ y = x \]

3. **Differentiate \( y \)**:
Now we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(x) = 1 \]

4. **Evaluate at \( x = \frac{\pi}{4} \)**:
Since \( \frac{dy}{dx} = 1 \) for all \( x \) in the interval \( [0, \pi] \), we can conclude that:
\[ \frac{dy}{dx} \bigg|_{x = \frac{\pi}{4}} = 1 \]

5. **Final Result**:
Thus, the value of \( \frac{dy}{dx} \) at \( x = \frac{\pi}{4} \) is:
\[ \frac{dy}{dx} = 1 \]

### Conclusion:
The correct answer is not explicitly listed in the options provided, but we have determined that \( \frac{dy}{dx} = 1 \).