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

Options

A. ` (-2)/( 1+x^(2))`
B. ` (2)/( 1+x^(2))`
C. ` (-1)/( 1+x^(2))`
D. ` (1)/( 1+x^(2))`

Answer

To solve the problem \( y = \cos^{-1}(\sqrt{1+x^2}) \) and find \( \frac{dy}{dx} \), we will use the chain rule of differentiation. Here’s a step-by-step solution:

### Step 1: Differentiate using the chain rule
We know that if \( y = \cos^{-1}(u) \), then:
\[ \frac{dy}{dx} = -\frac{1}{\sqrt{1 - u^2}} \cdot \frac{du}{dx} \]
where \( u = \sqrt{1 + x^2} \).

### Step 2: Find \( \frac{du}{dx} \)
Now, we need to differentiate \( u = \sqrt{1 + x^2} \):
\[ u = (1 + x^2)^{1/2} \]
Using the chain rule:
\[ \frac{du}{dx} = \frac{1}{2}(1 + x^2)^{-1/2} \cdot (2x) = \frac{x}{\sqrt{1 + x^2}} \]

### Step 3: Substitute \( u \) back into the derivative
Next, we substitute \( u \) back into the derivative formula:
\[ \frac{dy}{dx} = -\frac{1}{\sqrt{1 - (\sqrt{1 + x^2})^2}} \cdot \frac{x}{\sqrt{1 + x^2}} \]

### Step 4: Simplify the expression
Now, simplify \( 1 - (\sqrt{1 + x^2})^2 \):
\[ 1 - (1 + x^2) = -x^2 \]
Thus, we have:
\[ \frac{dy}{dx} = -\frac{1}{\sqrt{-x^2}} \cdot \frac{x}{\sqrt{1 + x^2}} \]
Since \( \sqrt{-x^2} = i|x| \) (where \( i \) is the imaginary unit), we can write:
\[ \frac{dy}{dx} = -\frac{1}{i|x|} \cdot \frac{x}{\sqrt{1 + x^2}} \]

### Step 5: Final simplification
Now, simplifying further:
\[ \frac{dy}{dx} = -\frac{x}{i|x|\sqrt{1 + x^2}} \]
If \( x \) is positive, \( |x| = x \), and if \( x \) is negative, \( |x| = -x \). Thus, we can write:
\[ \frac{dy}{dx} = -\frac{1}{i\sqrt{1 + x^2}} \]

### Final Answer
The final result for \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = -\frac{1}{i\sqrt{1 + x^2}} \]