If `y=(x+1)/(x-1)`, then what is the value of `(dy)/(dx)` ?

Options

A. `(-2)/(x-1)`
B. `(-2)/((x-1)^(2))`
C. `(2)/((x-1)^(2))`
D. `(2)/((x-1))`

Answer

To find the derivative of the function \( y = \frac{x + 1}{x - 1} \), we will use the quotient rule. The quotient rule states that if you have a function in the form of \( y = \frac{u}{v} \), then the derivative \( \frac{dy}{dx} \) is given by:

\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

where \( u = x + 1 \) and \( v = x - 1 \).

### Step 1: Identify \( u \) and \( v \)
Let:
- \( u = x + 1 \)
- \( v = x - 1 \)

### Step 2: Find \( \frac{du}{dx} \) and \( \frac{dv}{dx} \)
Now we need to find the derivatives of \( u \) and \( v \):
- \( \frac{du}{dx} = 1 \) (since the derivative of \( x + 1 \) is 1)
- \( \frac{dv}{dx} = 1 \) (since the derivative of \( x - 1 \) is 1)

### Step 3: Apply the Quotient Rule
Now we can apply the quotient rule:

\[ \frac{dy}{dx} = \frac{(x - 1)(1) - (x + 1)(1)}{(x - 1)^2} \]

### Step 4: Simplify the numerator
Now simplify the numerator:

\[ = \frac{x - 1 - (x + 1)}{(x - 1)^2} \]

Distributing the negative sign:

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

This simplifies to:

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

### Final Result
Thus, the derivative \( \frac{dy}{dx} \) is:

\[ \frac{dy}{dx} = \frac{-2}{(x - 1)^2} \]