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

Options

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

Answer

To find the derivative of the function \( y = \sin^{-1}(\sqrt{1 - x^2}) \), we will use the chain rule and the derivative of the inverse sine function. Here’s a step-by-step solution:

### Step 1: Identify the function
We have:
\[ y = \sin^{-1}(\sqrt{1 - x^2}) \]

### Step 2: Differentiate using the chain rule
The derivative of \( \sin^{-1}(u) \) with respect to \( x \) is given by:
\[ \frac{dy}{dx} = \frac{1}{\sqrt{1 - u^2}} \cdot \frac{du}{dx} \]
where \( u = \sqrt{1 - x^2} \).

### Step 3: Find \( u \) and its derivative
First, we need to find \( u \):
\[ u = \sqrt{1 - x^2} \]
Now, differentiate \( u \):
\[ \frac{du}{dx} = \frac{1}{2\sqrt{1 - x^2}} \cdot (-2x) = \frac{-x}{\sqrt{1 - x^2}} \]

### Step 4: Substitute \( u \) and \( \frac{du}{dx} \) into the derivative formula
Now substitute \( u \) and \( \frac{du}{dx} \) into the derivative formula:
\[ \frac{dy}{dx} = \frac{1}{\sqrt{1 - (\sqrt{1 - x^2})^2}} \cdot \frac{-x}{\sqrt{1 - x^2}} \]

### Step 5: Simplify the expression
We simplify \( 1 - (\sqrt{1 - x^2})^2 \):
\[ 1 - (\sqrt{1 - x^2})^2 = 1 - (1 - x^2) = x^2 \]
Thus, we have:
\[ \frac{dy}{dx} = \frac{1}{\sqrt{x^2}} \cdot \frac{-x}{\sqrt{1 - x^2}} \]
Since \( \sqrt{x^2} = |x| \), we can write:
\[ \frac{dy}{dx} = \frac{-x}{|x| \sqrt{1 - x^2}} \]

### Step 6: Final result
Assuming \( x \) is positive (as \( \sin^{-1} \) is defined for \( \sqrt{1 - x^2} \)), we can simplify further:
\[ \frac{dy}{dx} = \frac{-1}{\sqrt{1 - x^2}} \]

Thus, the final answer is:
\[ \frac{dy}{dx} = -\frac{1}{\sqrt{1 - x^2}} \]