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

Options

A. ` (sqrt( 1-y^(2)))/( 1+2y ^(2))`
B. ` (sqrt(1-y^(2)))/( 1-2y ^(2))`
C. ` 0`
D. `x`

Answer

To find \(\frac{dy}{dx}\) given the equation \(x = y \sqrt{1 - y^2}\), we can follow these steps:

### Step 1: Square both sides
Start by squaring both sides of the equation to eliminate the square root.

\[ x^2 = (y \sqrt{1 - y^2})^2 \]

This simplifies to:

\[ x^2 = y^2 (1 - y^2) \]

### Step 2: Expand the equation
Now, expand the right-hand side:

\[ x^2 = y^2 - y^4 \]

### Step 3: Rearrange the equation
Rearranging gives us:

\[ x^2 + y^4 - y^2 = 0 \]

### Step 4: Differentiate both sides
Now, differentiate both sides with respect to \(x\):

\[ \frac{d}{dx}(x^2) + \frac{d}{dx}(y^4) - \frac{d}{dx}(y^2) = 0 \]

Using the chain rule, we differentiate:

\[ 2x + 4y^3 \frac{dy}{dx} - 2y \frac{dy}{dx} = 0 \]

### Step 5: Factor out \(\frac{dy}{dx}\)
Now, we can factor out \(\frac{dy}{dx}\):

\[ 2x + (4y^3 - 2y) \frac{dy}{dx} = 0 \]

### Step 6: Solve for \(\frac{dy}{dx}\)
Rearranging gives:

\[ (4y^3 - 2y) \frac{dy}{dx} = -2x \]

Now, divide both sides by \((4y^3 - 2y)\):

\[ \frac{dy}{dx} = \frac{-2x}{4y^3 - 2y} \]

### Step 7: Simplify the expression
We can simplify this further:

\[ \frac{dy}{dx} = \frac{-x}{2y^3 - y} \]

### Step 8: Substitute for \(x/y\)
Recall that \(x = y \sqrt{1 - y^2}\), so:

\[ \frac{dy}{dx} = \frac{-y \sqrt{1 - y^2}}{2y^3 - y} \]

### Step 9: Final expression
Thus, we can express \(\frac{dy}{dx}\) as:

\[ \frac{dy}{dx} = \frac{\sqrt{1 - y^2}}{1 - 2y^2} \]