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

Options

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

Explanation

To find \(\frac{dy}{dx}\) given the equation \(x = \sqrt{1 - y^2}\), we will differentiate both sides with respect to \(x\). Here is the step-by-step solution:

### Step 1: Write the equation
We start with the equation:
\[ x = \sqrt{1 - y^2} \]

### Step 2: Differentiate both sides
We differentiate both sides with respect to \(x\). The left side differentiates to:
\[ \frac{d}{dx}(x) = 1 \]

For the right side, we apply the chain rule. The derivative of \(\sqrt{1 - y^2}\) is:
\[ \frac{d}{dx}(\sqrt{1 - y^2}) = \frac{1}{2\sqrt{1 - y^2}} \cdot \frac{d}{dx}(1 - y^2) \]
Now, we differentiate \(1 - y^2\):
\[ \frac{d}{dx}(1 - y^2) = 0 - 2y \frac{dy}{dx} = -2y \frac{dy}{dx} \]

Putting it all together, we have:
\[ \frac{d}{dx}(\sqrt{1 - y^2}) = \frac{1}{2\sqrt{1 - y^2}} \cdot (-2y \frac{dy}{dx}) = -\frac{y}{\sqrt{1 - y^2}} \frac{dy}{dx} \]

### Step 3: Set the derivatives equal
Now we set the derivatives equal to each other:
\[ 1 = -\frac{y}{\sqrt{1 - y^2}} \frac{dy}{dx} \]

### Step 4: Solve for \(\frac{dy}{dx}\)
To isolate \(\frac{dy}{dx}\), we rearrange the equation:
\[ \frac{dy}{dx} = -\frac{\sqrt{1 - y^2}}{y} \]

### Step 5: Simplify the expression
We can rewrite the expression:
\[ \frac{dy}{dx} = -\frac{\sqrt{1 - y^2}}{y} \]

### Final Answer
Thus, the final answer for \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = -\frac{\sqrt{1 - y^2}}{y} \]