If : `y=sqrt((1-x)(1+x))," then: "`

Options

A. `(1-x^(2))*(dy)/(dx)-xy=0`
B. `(1-x^(2))*(dy)/(dx)+xy=0`
C. `(1-x^(2))*(dy)/(dx)-2xy=0`
D. `(1-x^(2))*(dy)/(dx)+2xy=0`

Answer

To find the derivative of the function \( y = \sqrt{(1-x)(1+x)} \), we will follow these steps:

### Step 1: Rewrite the function
We start with the function:
\[ y = \sqrt{(1-x)(1+x)} \]
This can be simplified using the difference of squares:
\[ y = \sqrt{1 - x^2} \]

### Step 2: Differentiate using the chain rule
To differentiate \( y \), we will use the chain rule. The derivative of \( \sqrt{u} \) is \( \frac{1}{2\sqrt{u}} \cdot \frac{du}{dx} \). Here, \( u = 1 - x^2 \).

First, we find \( \frac{du}{dx} \):
\[ u = 1 - x^2 \implies \frac{du}{dx} = -2x \]

Now, applying the chain rule:
\[ \frac{dy}{dx} = \frac{1}{2\sqrt{1 - x^2}} \cdot (-2x) \]

### Step 3: Simplify the derivative
Now we simplify the expression:
\[ \frac{dy}{dx} = \frac{-2x}{2\sqrt{1 - x^2}} = \frac{-x}{\sqrt{1 - x^2}} \]

### Step 4: Write the final answer
Thus, the derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{-x}{\sqrt{1 - x^2}} \]

### Step 5: Formulate the differential equation
We know that \( y^2 = 1 - x^2 \). Therefore, we can express \( y^2 \) in terms of \( x \):
\[ y^2 = 1 - x^2 \implies x^2 = 1 - y^2 \]

Now, substituting \( y^2 \) back into our derivative:
\[ y^2 \frac{dy}{dx} + xy = 0 \]

This gives us the required differential equation.

### Final Answer
The final differential equation is:
\[ (1 - x^2) \frac{dy}{dx} + xy = 0 \]

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