Find ` (dy)/(dx) " when " y = sin^(-1) sqrt((1+x^(2))/2)`

Explanation

To find \(\frac{dy}{dx}\) when \(y = \sin^{-1}\left(\sqrt{\frac{1+x^2}{2}}\right)\), we will follow these steps:

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

### Step 2: Simplify the expression inside the inverse sine
To differentiate \(y\), we can simplify the expression inside the inverse sine. We can express \(\frac{1+x^2}{2}\) in a different form:
\[ \sqrt{\frac{1+x^2}{2}} = \sqrt{\frac{1}{2} + \frac{x^2}{2}} = \sqrt{\frac{1 + x^2}{2}} \]

### Step 3: Use trigonometric identities
Let \(x^2 = 2\cos(2\theta)\). Then, we can find \(\theta\) as:
\[ \cos(2\theta) = \frac{x^2}{2} \]
Thus, we have:
\[ \theta = \frac{1}{2} \cos^{-1}\left(\frac{x^2}{2}\right) \]

### Step 4: Substitute back into \(y\)
Now substituting back into \(y\):
\[ y = \sin^{-1}\left(\sqrt{\frac{1 + \cos(2\theta)}{2}}\right) \]
Using the identity \(1 + \cos(2\theta) = 2\cos^2(\theta)\):
\[ y = \sin^{-1}\left(\sqrt{\cos^2(\theta)}\right) = \sin^{-1}(\cos(\theta)) \]

### Step 5: Use the complementary angle identity
Using the identity \(\sin^{-1}(\cos(\theta)) = \frac{\pi}{2} - \theta\):
\[ y = \frac{\pi}{2} - \theta \]

### Step 6: Substitute \(\theta\) back
Substituting \(\theta\) back in:
\[ y = \frac{\pi}{2} - \frac{1}{2} \cos^{-1}\left(\frac{x^2}{2}\right) \]

### Step 7: Differentiate \(y\)
Now, we differentiate \(y\) with respect to \(x\):
\[ \frac{dy}{dx} = 0 - \frac{1}{2} \cdot \frac{d}{dx}\left(\cos^{-1}\left(\frac{x^2}{2}\right)\right) \]
Using the derivative of \(\cos^{-1}(u)\):
\[ \frac{d}{dx}\left(\cos^{-1}(u)\right) = -\frac{1}{\sqrt{1-u^2}} \cdot \frac{du}{dx} \]
where \(u = \frac{x^2}{2}\) and \(\frac{du}{dx} = x\).

### Step 8: Substitute \(u\) back
Now substituting \(u\) back:
\[ \frac{dy}{dx} = -\frac{1}{2} \cdot \left(-\frac{1}{\sqrt{1 - \left(\frac{x^2}{2}\right)^2}} \cdot x\right) \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{x}{2\sqrt{1 - \frac{x^4}{4}}} \]
which can be rewritten as:
\[ \frac{dy}{dx} = \frac{x}{\sqrt{4 - x^4}} \]

### Final Answer
Thus, the derivative is:
\[ \frac{dy}{dx} = \frac{x}{\sqrt{4 - x^4}} \]