If ` y=cos ^(-1) ((cos x+ sin x )/(sqrt( 2))),then (dy)/(dx)=`

Options

A. `2`
B. `-2`
C. `0`
D. `-1`

Answer

To solve the problem \( y = \cos^{-1} \left( \frac{\cos x + \sin x}{\sqrt{2}} \right) \) and find \( \frac{dy}{dx} \), we can follow these steps:

### Step 1: Simplify the expression for \( y \)

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

We can rewrite the numerator:
\[ \frac{\cos x + \sin x}{\sqrt{2}} = \frac{1}{\sqrt{2}} (\cos x + \sin x) \]

Using the angle addition formula for sine, we can express this as:
\[ \frac{1}{\sqrt{2}} (\cos x + \sin x) = \sin\left(x + \frac{\pi}{4}\right) \]
Thus, we have:
\[ y = \cos^{-1} \left( \sin\left(x + \frac{\pi}{4}\right) \right) \]

### Step 2: Differentiate \( y \)

To differentiate \( y \), we use the derivative of \( \cos^{-1}(u) \):
\[ \frac{dy}{dx} = -\frac{1}{\sqrt{1 - u^2}} \cdot \frac{du}{dx} \]
where \( u = \sin\left(x + \frac{\pi}{4}\right) \).

### Step 3: Find \( \frac{du}{dx} \)

Now we need to differentiate \( u \):
\[ u = \sin\left(x + \frac{\pi}{4}\right) \]
Using the chain rule:
\[ \frac{du}{dx} = \cos\left(x + \frac{\pi}{4}\right) \cdot \frac{d}{dx}(x + \frac{\pi}{4}) = \cos\left(x + \frac{\pi}{4}\right) \]

### Step 4: Substitute \( u \) into the derivative formula

Now we substitute \( u \) back into the derivative:
\[ \frac{dy}{dx} = -\frac{1}{\sqrt{1 - \sin^2\left(x + \frac{\pi}{4}\right)}} \cdot \cos\left(x + \frac{\pi}{4}\right) \]

### Step 5: Simplify the expression

Using the identity \( 1 - \sin^2\theta = \cos^2\theta \):
\[ \sqrt{1 - \sin^2\left(x + \frac{\pi}{4}\right)} = \cos\left(x + \frac{\pi}{4}\right) \]

Thus, we have:
\[ \frac{dy}{dx} = -\frac{\cos\left(x + \frac{\pi}{4}\right)}{\cos\left(x + \frac{\pi}{4}\right)} = -1 \]

### Final Answer

Therefore, the derivative is:
\[ \frac{dy}{dx} = -1 \]