Find `dy/dx` for each of the following
`y=((x-sqrtx)/(1-2x))^(2)`

Answer

To find \(\frac{dy}{dx}\) for the function

\[ y = \left(\frac{x - \sqrt{x}}{1 - 2x}\right)^2, \]

we will use the chain rule and the quotient rule.

### Step 1: Identify the inner function
Let
\[ u = \frac{x - \sqrt{x}}{1 - 2x}. \]
Then, we can rewrite \(y\) as:
\[ y = u^2. \]

### Step 2: Differentiate using the chain rule
Using the chain rule, we have:
\[ \frac{dy}{dx} = 2u \cdot \frac{du}{dx}. \]

### Step 3: Differentiate \(u\) using the quotient rule
To find \(\frac{du}{dx}\), we apply the quotient rule:
\[ \frac{du}{dx} = \frac{v \cdot \frac{d}{dx}(u) - u \cdot \frac{d}{dx}(v)}{v^2}, \]
where \(u = x - \sqrt{x}\) and \(v = 1 - 2x\).

#### Step 3.1: Differentiate the numerator
The numerator \(u = x - \sqrt{x}\) can be differentiated as follows:
\[ \frac{d}{dx}(x - \sqrt{x}) = 1 - \frac{1}{2\sqrt{x}}. \]

#### Step 3.2: Differentiate the denominator
The denominator \(v = 1 - 2x\) can be differentiated as:
\[ \frac{d}{dx}(1 - 2x) = -2. \]

### Step 4: Substitute into the quotient rule
Now substituting back into the quotient rule:
\[ \frac{du}{dx} = \frac{(1 - 2x)\left(1 - \frac{1}{2\sqrt{x}}\right) - (x - \sqrt{x})(-2)}{(1 - 2x)^2}. \]

### Step 5: Simplify the expression
Now we simplify the numerator:
1. Expand the first term:
\[ (1 - 2x)\left(1 - \frac{1}{2\sqrt{x}}\right) = 1 - \frac{1}{2\sqrt{x}} - 2x + \frac{x}{\sqrt{x}} = 1 - 2x - \frac{1}{2\sqrt{x}} + \sqrt{x}. \]
2. The second term becomes:
\[ 2(x - \sqrt{x}) = 2x - 2\sqrt{x}. \]
3. Combine the two parts:
\[ 1 - 2x - \frac{1}{2\sqrt{x}} + \sqrt{x} + 2x - 2\sqrt{x} = 1 - \frac{1}{2\sqrt{x}} - \sqrt{x}. \]

### Step 6: Final expression for \(\frac{dy}{dx}\)
Now substituting back into the expression for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = 2u \cdot \frac{1 - \frac{1}{2\sqrt{x}} - \sqrt{x}}{(1 - 2x)^2}. \]
Substituting \(u\) back:
\[ \frac{dy}{dx} = 2\left(\frac{x - \sqrt{x}}{1 - 2x}\right)\cdot \frac{1 - \frac{1}{2\sqrt{x}} - \sqrt{x}}{(1 - 2x)^2}. \]

### Final Answer
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{2(x - \sqrt{x})\left(1 - \frac{1}{2\sqrt{x}} - \sqrt{x}\right)}{(1 - 2x)^3}. \]