If ` x+sqrt( xy)+ y=1 ,then (dy)/(dx)=`

Options

A. ` (2sqrt( xy)+x)/( 2sqrt(xy) +y)`
B. ` (2sqrt( xy)+y)/( 2sqrt(xy) +x)`
C. ` -((2sqrt( xy)+x)/( 2sqrt(xy) +y))`
D. ` -((2sqrt( xy)+y)/( 2sqrt(xy) +x))`

Answer

To find \(\frac{dy}{dx}\) for the equation \(x + \sqrt{xy} + y = 1\), we will differentiate both sides of the equation with respect to \(x\). Let's go through the steps systematically.

### Step 1: Rearranging the Equation
We start with the equation:
\[ x + \sqrt{xy} + y = 1 \]
We can rearrange it for easier differentiation:
\[ \sqrt{xy} = 1 - x - y \]

### Step 2: Differentiate Both Sides
Now we differentiate both sides with respect to \(x\):
\[ \frac{d}{dx}(x) + \frac{d}{dx}(\sqrt{xy}) + \frac{d}{dx}(y) = \frac{d}{dx}(1) \]
The derivative of \(1\) is \(0\). So we have:
\[ 1 + \frac{d}{dx}(\sqrt{xy}) + \frac{dy}{dx} = 0 \]

### Step 3: Differentiate \(\sqrt{xy}\)
To differentiate \(\sqrt{xy}\), we use the product rule. Let \(u = x\) and \(v = y\):
\[ \frac{d}{dx}(\sqrt{xy}) = \frac{1}{2\sqrt{xy}} \cdot \frac{d}{dx}(xy) \]
Using the product rule on \(xy\):
\[ \frac{d}{dx}(xy) = x\frac{dy}{dx} + y \]
Thus,
\[ \frac{d}{dx}(\sqrt{xy}) = \frac{1}{2\sqrt{xy}}(x\frac{dy}{dx} + y) \]

### Step 4: Substitute Back
Now substitute back into our differentiated equation:
\[ 1 + \frac{1}{2\sqrt{xy}}(x\frac{dy}{dx} + y) + \frac{dy}{dx} = 0 \]

### Step 5: Isolate \(\frac{dy}{dx}\)
Combine the terms involving \(\frac{dy}{dx}\):
\[ 1 + \frac{y}{2\sqrt{xy}} + \left(\frac{x}{2\sqrt{xy}} + 1\right)\frac{dy}{dx} = 0 \]
Rearranging gives:
\[ \left(\frac{x}{2\sqrt{xy}} + 1\right)\frac{dy}{dx} = -\left(1 + \frac{y}{2\sqrt{xy}}\right) \]
Now, isolate \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = -\frac{1 + \frac{y}{2\sqrt{xy}}}{\frac{x}{2\sqrt{xy}} + 1} \]

### Step 6: Simplifying the Expression
To simplify, multiply the numerator and denominator by \(2\sqrt{xy}\):
\[ \frac{dy}{dx} = -\frac{2\sqrt{xy} + y}{x + 2\sqrt{xy}} \]

### Final Result
Thus, the final expression for \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = -\frac{2\sqrt{xy} + y}{x + 2\sqrt{xy}} \]