Find `dy/dx if f(x)= 2/ (1-x)`

Answer

To find \(\frac{dy}{dx}\) for the function \(f(x) = \frac{2}{1-x}\), we will use the quotient rule of differentiation.

### Step-by-Step Solution:

1. **Identify the function**:
\[ f(x) = \frac{2}{1-x} \]
Here, we can identify \(u = 2\) (the numerator) and \(v = 1 - x\) (the denominator).

2. **Apply the Quotient Rule**:
The quotient rule states that if \(f(x) = \frac{u}{v}\), then:
\[ f'(x) = \frac{u'v - uv'}{v^2} \]
where \(u' = \frac{du}{dx}\) and \(v' = \frac{dv}{dx}\).

3. **Differentiate \(u\) and \(v\)**:
- Since \(u = 2\), we have:
\[ u' = 0 \]
- For \(v = 1 - x\), we differentiate:
\[ v' = -1 \]

4. **Substitute into the Quotient Rule**:
Now, substituting \(u\), \(u'\), \(v\), and \(v'\) into the quotient rule formula:
\[ f'(x) = \frac{(0)(1-x) - (2)(-1)}{(1-x)^2} \]

5. **Simplify the expression**:
\[ f'(x) = \frac{0 + 2}{(1-x)^2} = \frac{2}{(1-x)^2} \]

6. **Final Result**:
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{2}{(1-x)^2} \]