The derivative of `cos^(-1)((sinx+cosx)/(sqrt(2))),-(pi)/(4)ltxlt(pi)/(4)`, w.r.t.x is

Options

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

Answer

To find the derivative of \( y = \cos^{-1}\left(\frac{\sin x + \cos x}{\sqrt{2}}\right) \) with respect to \( x \), where \( -\frac{\pi}{4} < x < \frac{\pi}{4} \), we can follow these steps:

### Step 1: Rewrite the function
We start with the function:
\[ y = \cos^{-1}\left(\frac{\sin x + \cos x}{\sqrt{2}}\right) \]

### Step 2: Simplify the argument of the inverse cosine
Notice that:
\[ \sin x + \cos x = \sqrt{2} \left(\sin x \cdot \frac{1}{\sqrt{2}} + \cos x \cdot \frac{1}{\sqrt{2}}\right) = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right) \]
Thus, we can rewrite:
\[ \frac{\sin x + \cos x}{\sqrt{2}} = \sin\left(x + \frac{\pi}{4}\right) \]
So, we have:
\[ y = \cos^{-1}\left(\sin\left(x + \frac{\pi}{4}\right)\right) \]

### Step 3: Use the identity for cosine inverse of sine
Using the identity \( \cos^{-1}(\sin \theta) = \frac{\pi}{2} - \theta \) for \( \theta \) in the appropriate range, we can write:
\[ y = \frac{\pi}{2} - \left(x + \frac{\pi}{4}\right) = \frac{\pi}{2} - x - \frac{\pi}{4} = \frac{\pi}{4} - x \]

### Step 4: Differentiate with respect to \( x \)
Now, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}\left(\frac{\pi}{4} - x\right) = -1 \]

### Final Answer
Thus, the derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = -1 \]