If `y=cos^(-1) (cos x),"then " (dy)/(dx)` is

Options

A. 1 and the `2^(nd) ` and `3^(rd )` quadrants of the plane
B. -1 and the `3^(rd) ` and `4^(th )` quadrants of the plane
C. 1 in the whole plane
D. `-1` in the whole plane I152

Answer

To solve the problem where \( y = \cos^{-1}(\cos x) \), we need to find the derivative \( \frac{dy}{dx} \).

### Step-by-Step Solution:

1. **Understanding the Function**:
The function \( y = \cos^{-1}(\cos x) \) can be simplified based on the properties of the inverse cosine function. The output of \( \cos^{-1}(z) \) is defined in the range \( [0, \pi] \). Therefore, \( y \) will equal \( x \) if \( x \) is in the interval \( [0, \pi] \) and will equal \( 2\pi - x \) if \( x \) is in the interval \( [\pi, 2\pi] \).

2. **Finding the Derivative**:
To differentiate \( y \) with respect to \( x \), we will consider the derivative of \( \cos^{-1}(u) \), where \( u = \cos x \). The derivative of \( \cos^{-1}(u) \) is given by:
\[ \frac{dy}{du} = -\frac{1}{\sqrt{1 - u^2}} \]
Since \( u = \cos x \), we have:
\[ \frac{du}{dx} = -\sin x \]

3. **Applying the Chain Rule**:
Using the chain rule, we can find \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = -\frac{1}{\sqrt{1 - \cos^2 x}} \cdot (-\sin x) \]

4. **Simplifying the Expression**:
We know that \( 1 - \cos^2 x = \sin^2 x \). Therefore, we can substitute this into our expression:
\[ \frac{dy}{dx} = \frac{\sin x}{\sqrt{\sin^2 x}} = \frac{\sin x}{|\sin x|} \]
The absolute value \( |\sin x| \) will depend on the interval of \( x \). However, for the intervals \( [0, \pi] \) and \( [\pi, 2\pi] \), we can conclude:
- For \( x \in [0, \pi] \), \( \sin x \geq 0 \) so \( |\sin x| = \sin x \).
- For \( x \in [\pi, 2\pi] \), \( \sin x \leq 0 \) so \( |\sin x| = -\sin x \).

5. **Final Result**:
Thus, in the interval \( [0, \pi] \), \( \frac{dy}{dx} = 1 \) and in the interval \( [\pi, 2\pi] \), \( \frac{dy}{dx} = -1 \). However, since the question does not specify the interval, we can conclude:
\[ \frac{dy}{dx} = 1 \text{ (for } x \in [0, \pi] \text{)} \]

### Final Answer:
\[ \frac{dy}{dx} = 1 \]